NBER WORKING PAPER SERIES

THE EMERGENCE OF WEAK, DESPOTIC AND INCLUSIVE STATES

Daron AcemogluJames A. Robinson

Working Paper 23657http://www.nber.org/papers/w23657

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138August 2017

We thank Pooya Molavi for exceptional research assistance, Marco Battaglini, Roland Benabou, Kim Hill, Josh Ober and Mark Pyzyk for discussions and seminar participants at the ASSA 2017, CIFAR, Chicago, NBER, University of Illinois, Lund, Northwestern and Yale for useful comments and suggestions. Daron Acemoglu gratefully acknowledges financial support from the Carnegie Foundation and ARO MURI Award No. W911NF-12-1-0509. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

© 2017 by Daron Acemoglu and James A. Robinson. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.

The Emergence of Weak, Despotic and Inclusive StatesDaron Acemoglu and James A. RobinsonNBER Working Paper No. 23657August 2017JEL No. H4,H7,P16

ABSTRACT

Societies under similar geographic and economic conditions and subject to similar external influences nonetheless develop very different types of states. At one extreme are weak states with little capacity and ability to regulate economic or social relations. At the other are despotic states which dominate civil society. Yet there are others which are locked into an ongoing competition with civil society and it is these, not the despotic ones, that develop the greatest capacity. We develop a dynamic contest model of the potential competition between state (controlled by a ruler or a group of elites) and civil society (representing non-elite citizens), where both players can invest to increase their power. The model leads to different types of steady states depending on initial conditions. One type of steady state, corresponding to a weak state, emerges when civil society is strong relative to the state (e.g., having developed social norms limiting political hierarchy). Another type of steady state, corresponding to a despotic state, originates from initial conditions where the state is powerful and civil society is weak. A third type of steady state, which we refer to as an inclusive state, emerges when state and civil society are more evenly matched. In this case, each party has greater incentives to invest to keep up with the other, and this leads to the most powerful and capable type of state, while simultaneously incentivizing civil society to be equally powerful as well. Our framework highlights that comparative statics with respect to structural factors such as geography, economic conditions or external threats, are conditional — in the sense that depending on initial conditions they can shift a society into or out of the basin of attraction of the inclusive state.

Daron AcemogluDepartment of Economics, E52-446MIT77 Massachusetts AvenueCambridge, MA 02139and CIFARand also NBERdaron@mit.edu

James A. RobinsonUniversity of ChicagoHarris School of Public Policy1155 East 60th StreetChicago, Illinois 60637and NBERjamesrobinson@uchicago.edu

1 Introduction

The capacity of the state to enforce laws, provide public services, and regulate and tax economic

activity varies enormously across countries. The dominant paradigm in social science to explain

the development of state capacity links this diversity to the ability of a powerful group, elite or

charismatic leader to dominate other powerful actors in society and build institutions such as a fiscal

system or bureaucracy (e.g., Huntington, 1968). This paradigm also relates this ability to certain

structural factors such as geography, ecology, natural resources and population density (Mahdavy,

1970, Diamond, 1997, Herbst, 2000, Fukuyama, 2011, 2014), the threat of war (Hintze, 1975, Brewer,

1989, Tilly, 1975, 1990, Besley and Persson, 2009, 2011, O’Brien, 2011, Gennaioli and Voth, 2015),

or the nature of economic activity (Mann, 1986, 1993, Acemoglu, 2005, Spruyt, 2009, Besley and

Persson, 2011, Mayshar, Moav and Neeman, 2011). However, historically, societies with similar

ecologies, geographies, initial economic structures and external threats have diverged sharply in terms

of the development of their states. Moreover, in many instances in which the state has built capacity,

it has not dominated a meek society; on the contrary, it has had to continuously contend and struggle

with a strong, assertive (civil) society.1

These dynamics are illustrated by the historical evolution of the power of the state and state-

society relations in Europe. European nations share not only a great deal of history and culture but

also broadly similar economic conditions. And yet, the types of states and political dynamics we

observe in the continent over the last several centuries are hugely varied. At one end of the spectrum,

Prussia in the 19th century constructed an autocratic, militarized state under an absolutist monarchy,

backed by a traditional landowning Junker class, which continued to exercise enough authority to help

derail the Weimar democracy in the 1920s (Evans, 2005). Meanwhile, just to its south, the Swiss state

attained its final institutionalization in 1848, not as a consequence of an absolutist monarchy, but

from the bottom-up construction of independent republican cantons based on rural-urban communes.

A little further south, places in the Balkans, such as Montenegro, never had centralized state authority

at all. Prior to 1852 Montenegro was in effect a theocracy, but its ruling Bishop, the Vladika, could

exercise no coercive authority over the clans which dominated the society partly via a complex web

of traditions and social norms. As a consequence of this lack of state authority, blood feuds and other

inter-clan conflicts were extremely common.2

1Throughout, we use “civil society” and “society” interchangeably when this will cause no confusion.2Simic (1967) and Boehm (1986) emphasize the importance of clans and traditions as a constraint on state power

in Montenegro (see also Djilas, 1966). For example, “Continued attempts to impose centralized government were inconflict with tribal loyalty” (Simic, 1967, p. 87), and “It was only when their central leader attempted to institutionalizeforcible means of controlling feuds that the tribesmen stood firm in their right to follow their ancient traditions. Thiswas because they perceived in such interference a threat to their basic political autonomy.” (Boehm, 1986, p. 186).

Indeed, the first attempt at a codified law in 1796 by Vladika Peter I reflected the fact that order in society wasregulated by the institution of blood feuds. It included the clauses: “A man who strikes another with his hand, foot, orchibouk, shall pay him a fine of fifty sequins. If the man struck at once kills his aggressor, he shall not be punished. Norshall a man be punished for killing a thief caught in the act...” and “If a Montenegrin in self-defense kills a man whohas insulted him ... it shall be considered that the killing was involuntary.” (quoted in Durham, 1928, pp. 78-88). TheMontenegrin politician and writer Milovan Djilas describes the importance of blood feuds in the 1950s thus “the men ofseveral generations have died at the hands of Montenegrins, men of the same faith and name. My father’s grandfather,my own two grandfathers, my father, and my uncle were killed, as though a dread curse lay upon them . . . generationafter generation, and the bloody chain was not broken. The inherited fear and hatred of feuding clans was mightierthan fear and hatred of the enemy, the Turks. It seems to me that I was born with blood in my eyes. My first sight wasof blood. My first words were blood and bathed in blood” (1958, pp. 3-4).

1

This diversity is hard to explain based on the structural differences. Switzerland and Montenegro

are both mountainous (which Braudel, 1966, emphasized as crucial), were both part of the Roman

Empire, have been Christian for centuries, were specialized in similar economic activities such as

herding, and have been involved in continuous wars against external foes. Before the founding of

the Swiss Confederation in 1291, feuding was also common in that area. Scott, for example, notes:

“There is general agreement amongst recent historians that the origins of the Swiss Confederation

lay in the search for public order. The provisions of the Bundesbrief of 1291 were clearly directed

against feuding in the inner cantons” (1995, p. 98; see also Blickel, 1992). The parallels between

Switzerland and Prussia are even stronger. Both countries have very close cultural and ethnic roots

(and historically Switzerland had been settled by Germanic tribes, particularly the Alemanni), and

have shared similar religious identities before and after the Reformation. Though Prussia, unlike

Switzerland, was not part of the Holy Roman Empire, its institutions have been heavily influenced

by those of the Empire and had feudal roots similar to those of Switzerland.

In this paper, we develop a simple theory of state-society relations where the competition and

conflict between state and (civil) society is the main driver of the institutional change and the emer-

gence of state capacity (Acemoglu and Robinson, 2012). Small differences, such as those between

Prussia, Switzerland and Montenegro which we further discuss below, can set off political dynamics

in very different directions. In our model, though the elites that control the state wish to establish

dominance over society, the ability of society to develop its own strengths (in the form of coordination,

social norms and local organization) is central, because it induces the state to become even stronger

in order to compete with society. Likewise, the race against the state encourages society to invest

further in its capacity. When this balance between state and society is not achieved, either the state

fully dominates society or society is powerful and the state remains weak. Crucially, however, when

society is weak, state capacity is relatively limited as well, because the state can control society easily

and does not need to invest much in its own capacity (strength).3 In addition to highlighting the

vital role of the race between state and society in the development of state capacity, our theory shows

that, just as in the examples discussed above, polities with similar initial conditions and subject to

similar structural influences can nonetheless experience divergent state-society relations and evolution

of state capacity, because they may fall into the basins of attraction of different dynamic equilibria.

This history-dependent development of state capacity and our main theoretical results are summa-

rized in Figure 1. This figure plots the global dynamics of state-society relations. Region I illustrates

the Huntingtonian path, approximating the political dynamics of Prussia. Here, the state is stronger

than civil society to start with and fully dominates it; for this reason, we call this type of state

despotic. Region III is the case in which the social norms of the society, especially in how they are

able to act collectively and control political power and hierarchy, are strong and this prevents the

emergence of a powerful state, paving the way to weak states as in Montenegro. Region II illustrates

the happy middle ground where state and society are initially in balance, and this triggers a positive

competition between the two, whereby they both become stronger over time. We refer to this type of

3Throughout, we use power, strength and capacity interchangeably. In practice, one might wish to distinguishbetween the underlying, infrastructural power of the state, which then creates capacity to achieve certain goals orimplement certain objectives, but in our abstract model, this distinction does not arise.

2

Power of Society

Power of the State

Despotic State:Prussia,China

Weak State:Montenegro,

Somalia

Inclusive State:UK,

Switzerland

Region I Region II

Region III

Figure 1: The emergence and dynamics of weak, despotic and inclusive states.

state, which best resembles the Swiss path, as the inclusive state.4 As the figure shows, it is in this

inclusive case that the state achieves the greatest capacity. The fact that the capacity of the state

is greater in this case than in Region I highlights that it is the competition between state and civil

society that triggers greater investments by the state (or the ruler and elites controlling it)

Our theory and Figure 1 suggest that divergent political paths such as those between Prussia,

Switzerland and Montenegro may lie not in large differences in structural factors, but in small dif-

ferences that get amplified as a result of the competition between state and society. Such small

differences indeed favored the development of a powerful state in Prussia, which emerged out of the

militarized state of the Teutonic Knights to the east of the River Elbe, where feudalism was possi-

bly the most intense in Europe (Gerschenkron, 1943, Moore, 1966, Clark, 2009). In contrast, they

likely favored the weak state path in Montenegro, where the ‘herdsman’ culture was very strong and

a legitimate political order like the Holy Roman Empire was absent for several centuries. In con-

tradistinction to both of these cases, the powers of state and society were more evenly balanced in

Switzerland. Differently from Prussia, Swiss peasants were more ‘free’ (Steinberg, 2015), independent

cities such as Basel, Bern and Zurich played a more important economic and political role, and the

major demographic changes of the 14th century, in particular the Black Death of the 1340s, appear to

have weakened the elites even further (e.g., Morerod and Favrod, 2014). Compared to Montenegro,

4This terminology is motivated by the fact that, in this case, the state is not just strong (capable) but also evenlymatched with civil society, which is then able to actively participate in political conflict and partially check the domi-nation of the state and the elites that control it.

3

Switzerland’s history of established political order under the Holy Roman Empire and of corporations

such as monasteries and cathedral chapters (Church and Head, 2013, Morerod and Favrod, 2014) may

have created the small differences facilitating the emergence of a state capable of competing against

civil society.

Theoretically, our setup is one of a dynamic contest between two players, the elite controlling

the state and the (civil) society representing non-elite citizens. At each date, the state and society

both choose investments in their strength, and these strengths determine both the overall output in

the economy which is distributed between the elite controlling the state and the rest of the citizens,

and how this distribution takes place. We introduce some degree of economies of scale in the contest

technology so that the cost of investment for either state or society becomes higher if their strength

falls below a certain level.5 The interplay of contest incentives and the presence of economies of

scale underpins the emergence of three stable steady states as shown in Figure 1: when one party is

significantly stronger than the other, the weaker player is discouraged from investing. But since as

in contest models, part of the reason why each player invests is to be stronger than the other, the

discouragement of the weaker party also reduces the investment incentives of the dominant player.

In contrast, when the two players are evenly matched, they are both induced to invest more. These

results are an application of Harris and Vickers’ (1984, 1987) discouragement effect.6 After illustrating

the workings of our model in the simplest possible case where the players act myopically, we show

that similar results obtain when the players are forward-looking but sufficiently impatient.

Though our theory does not link the evolution of state capacity to structural factors, our general

framework provides comparative static results that clarify when a society is more likely to be in the

basin of attraction of different types of steady states (as already hinted by Figure 1). Specifically,

changes in underlying parameters — corresponding to technologies, economic conditions and the

external environment — change the basins of attraction of the three steady states. For example,

starting from the basin of attraction of Region III, a need for greater coordination in the economy,

resulting either because of increasing importance of public goods or national defense, can shift a

society into Region II, and thus trigger the long-run development of the state. However, crucially,

our framework also clarifies that the effects of structural factors are conditional — the same change

in external threats can also shift a society from Region II to Region I, thus triggering the evolution

of a despotic state and ultimately limiting the growth of state capacity. This, for example, explains

why theories such as those of Tilly mentioned above, which emphasize the threat of war as a driver

of state building, tend to have only limited explanatory power (see Pincus and Robinson, 2016, for a

historical discussion).

Our model is purposefully reduced-form and simple, which helps to highlight the potential gen-

erality of the forces we are emphasizing. One drawback of this strategy is that the mapping from

the investments by the state and society in our model to the data is not as clear as it would have

5This appears to be a reasonable assumption both for society and the state. Many scholars have argued that thereare increasing returns to collective action (e.g., Marwell and Oliver, 1993, Pearson, 2000) or the acquisition of socialcapital (Francois, 2002). It also appears reasonable that there are fixed costs involved in the creation of fiscal systemsor bureaucracies (e.g. Dharmapalaa, Slemrod and Wilson, 2011, Gauthier, 2013).

6See Dechenaux, Kovenock and Sheremeta (2015) for a survey of experimental evidence on discouragement effect incontests. See also Aghion, Bloom, Blundell, Griffith and Howitt (2005) and Aghion and Griffith (2008) for evidence onthe discouragement effect in the context of innovation investments.

4

been under a setup with explicit microstructure. We attempt to partially rectify this in Section 7

by revisiting the evolution of different types of states in Ancient Greece through the lenses of our

framework, and clarify the empirical content of the investments and the political economy forces in

our model. Specifically, this example, as well as several others from stateless societies studied by

anthropologists and discussed in Flannery and Marcus (2014) and Acemoglu and Robinson (2018),

show that many societies invest in and develop a whole range of norms and practices in order to

limit political hierarchy and the capability of powerful agents to impose their wishes. These norms

and practices in turn discourage the development of state capacity when initial political hierarchy is

limited and state institutions are weak, but can morph into institutional constraints on the exercise

of state power when initial state institutions are more developed as in the case of Athens.

Our paper is related to a number of literatures. As already discussed above, prominent in the

social science literature on state building are approaches that situate the roots of state capacity in

the ability of the state and groups controlling it to dominate society.7 In addition, these approaches

also emphasize the role of structural factors in triggering or preventing state building. Our theory

thus sharply differs from these approaches, and has much more in common with a few works in

sociology and political science emphasizing the interaction of state and society. Most importantly,

Migdal (1988, 2001) argues that weak states are a consequence of a strong society (as in our Region

III). Scott (2010) has similarly stressed the ability of people to resist the state and its interference.

Putnam (1993) argued that a strong society leads to better governance and bureaucratic effectiveness.

None of these scholars note our key distinction from the previous literature — the idea that state

capacity develops most strongly when state and civil society are matched in terms of their strengths

and compete dynamically.8 Acemoglu (2005) argues that the capacity of the state is highest when

it is “consensually strong”. In his model this emerges not because of competition between state and

society, but as a result of a repeated game equilibrium in which citizens are expected to replace rulers

who do not provide sufficient public goods or otherwise misbehave.

Our work is also related to a large literature in archaeology focusing on how societies start the

process of state formation (so-called “pristine state formation”). Most of these, for example Flannery

(1999) or Flannery and Marcus (2013), emphasize a ‘top-down’ elite centric approach, but other work,

particularly by Blanton and Fargher (2008), has placed equal weight on the role of society.

Finally, our model is an example of a dynamic contest, though most of our analysis involves myopic

players. Static models of contests in economics go back at least to Tullock (1980), and have been

7In addition to the works such as Huntington (1968), Tilly (1990) and Fukuyama (2011) mentioned above, thisincludes authors emphasizing the role of state capacity in enabling elites controlling the state to dominate society viavarious means, including repression (e.g., Anderson, 1974, Hechter and Brustein, 1980, Slater, 2010, Saylor, 2014).

The recent economics literature, mentioned above, mirrors these approaches. For example, Besley and Persson (2009,2011) focus on the incentives of the elites controlling the state to undertake investments to build state capacity and linkthis to the probability that they will lose power domestically and to external threats. Gennaioli and Voth (2015) developa model of the interaction between warfare and state capacity, while Mayshar, Moav and Neeman (2011) emphasizethe effect of the type of crop on state building. Other work by Acemoglu, Robinson and Santos-Villagran (2013) andAcemoglu, Ticchi and Vindigni (2011) again emphasize elite incentives.

8Our theory is also related to a few works stressing the implications of state centralization on civil society’s organi-zation. These latter works include Tilly (1995), who illustrates these political dynamics using the British case in the18th and 19th centuries, Acemoglu, Robinson and Torvik (2016), who develop a formal model along these lines, andHabermas (1989), who suggests that the origins of the “public sphere”, which can be viewed as an important aspect ofstrong society, lie in the process of state formation.

5

more systematically studied by Dixit (1987), Skaperdas (1992, 1996), Cornes and Hartley (2005),

and Corchon (2007). They are similar to models of (patent) races as in Loury (1979), and to all-pay

auctions as studied, among others, by Baye et al. (1996), Krishna and Morgan (1997) and Siegel

(2009). Our formulation uses a contest function in differences, introduced by Hirshleifer (1989), and

is mathematically closer to all-pay auctions (e.g., Che and Gale, 2000).9 Dynamic contests and related

racing models are more challenging and various special cases have been discussed in Fudenberg et al.

(1983), Harris and Vickers (1985, 1987), Grossman and Shapiro (1987), Grossman and Kim (1996),

Konrad (2009, 2012), and Cao (2014), while the literature on dynamic public good games deals

with related problems, but crucially without the contest element (e.g., Levhari and Mirman, 1980,

Lockwood and Thomas, 2002, and Battaglini, Nunnari and Palfrey, 2014). None of these papers, and

to the best of our knowledge no others, derive the possibility of three locally stable steady states with

different configurations of power.10

The rest of the paper is organized as follows. In the next section, we introduce our main model.

In Section 3, we characterize the dynamic equilibrium and steady states of this model when players

are short-lived or myopic. To maximize transparency, this section uses a number of simplifying

assumptions, many of which are relaxed later. Section 4 analyzes the same model with forward-

looking players, and establishes that the same results when these players are sufficiently impatient.

Section 5 relaxes one of the most important simplifying assumptions, allowing the investments of the

state and civil society to also affect the size of the pie to be divided. In this setup, it also provides

additional comparative static results on how different steady states and their basins of attraction

are affected by changes in parameters. Section 7 shows how our conceptual framework is useful

for interpreting the divergent paths of state capacity and state-society relations in Ancient Greece.

Finally, Section 8 concludes, while the Appendix provides some generalizations and microfoundations

for the setup studied in the main text.

2 Basic Model

In this section, we introduce our basic model, aimed at capturing the dynamics of conflict between

state and society discussed in the Introduction. We consider the state to be controlled by a ruler or

group of elites acting in a coordinated manner — motivating our convention of using elite and state

interchangeably in this paper. The main decision for the elite will be how much to invest in the power

of the state, which captures, among other things, the military power, the presence of state employees

(i.e., what Mann, 1986, refers to as the “infrastructural power of the state”) and the ability of the

state to regulate and tax economic activity. On the other side, we will consider groups interacting

with the state that do not have direct control over its actions. These groups could be other (e.g., local)

elites or regular citizens. Throughout, we will refer to them as “civil society” or simply as “society”.

Civil society will also invest in its power, partly as a defensive measure, to balance the power of

the elites and the state. As already discussed in the Introduction, these investments correspond to

9We show in the Appendix that the particular formulation is not critical for the results since the discouragementeffect arises in many standard models.

10In a very different context, Benabou, Ticchi and Vindigni (2015) emphasize how the initial balance of power betweenreligion and science can influence the long-run evolution of a society.

6

society’s efforts to coordinate its activities, its local organization and social norms that are useful for

limiting political hierarchy and the capability of powerful agents and the state to impose their wishes

on non-elite agents.

In this section we start with our general framework. We then analyze the cases in which both state

and civil society are myopic (e.g., consisting of non-overlapping generations) and fully forward-looking

separately. The framework we present here is reduced form. The Appendix outlines a model that

is more explicit about the economic and conflict decisions that civil society takes, and shows that it

maps into our reduced-form model.

2.1 Preferences and Conflict

We start with a discrete time setup, where period length is ∆ > 0 and will later be taken to be small,

so that we work with differential rather than difference equations in characterizing the dynamics. At

time t, the state variables inherited from the previous period are (xt−∆, st−∆) ∈ [0, 1]2, where the

first element corresponds to the strength of civil society and the second to the strength of the state

controlled by the elite.

At each point in time, the elite or the state is represented by a single player, and civil society is also

represented by a single player. In the next two sections, we study both the case in which these players

are short-lived and are immediately replaced by another player (so that we have a non-overlapping

generations model with “myopic” players), and the case in which players are long-lived and maximize

their discounted sum of utilities.

At time t, players simultaneously choose their investments, ixt ≥ 0 and ist ≥ 0, which determine

their current strengths according to the equations:

xt = xt−∆ + ixt ∆− δ∆, (1)

and

st = st−∆ + ist∆− δ∆, (2)

where δ > 0 is the depreciation of the strength of both parties between periods. Both investment and

depreciation are multiplied by the period length, ∆, since they represent “flow” variables, and when

period length is taken to be small, they will be suitably downscaled.11

The cost of investment for civil society during a period of length ∆ is given as ∆ · Cx(ixt , xt−∆)

where

Cx(ixt , xt−∆) =

{cx(ixt ) if xt−∆ > γx,

cx(ixt ) + (γx − xt−∆) ixt if xt−∆ ≤ γx.

This cost function is multiplied by ∆, since it is the cost of investing an amount ixt during the period

of length ∆ (as captured by equation (1)). The presence of the term γx > 0, on the other hand,

captures the “increasing returns” nature of conflict mentioned in the Introduction: starting from a

low level of conflict capacity, it is more costly to build up this capacity. We specify this in a very

simple form here, with the cost of investments increasing linearly as last period’s conflict capacity

11Assuming that depreciation is independent of the current level of the strength of the state or civil society is forconvenience only. In addition, we can easily allow the two state variables to have different depreciation rates, but donot do so in order to prevent the notation from becoming more cumbersome.

7

falls below the threshold γx. This increasing returns aspect plays an important role in our analysis

as we emphasize below.

The cost of investment for the state during a period of length ∆ is similarly given as ∆·Cs(ist , st−∆)

where

Cs(ist , st−∆) =

{cs(i

st ) if st−∆ > γs,

cs(ist ) + (γs − st−∆)ist if st−∆ ≤ γs.

In these expressions, it will often be more convenient to eliminate investment levels and directly

work with the two state variables, xt and st, especially when we take ∆ to be small and transition

to continuous time. In preparation for this transition, let us substitute out the investment levels and

observe that the cost function for civil society and state can be written as:

Cx(xt, xt−∆) = cx

(xt − xt−∆

∆+ δ

)+ max {γx − xt−∆, 0}

(xt − xt−∆

∆+ δ

),

and

Cs(st, st−∆) = cs

(st − st−∆

∆+ δ

)+ max {γs − st−∆, 0}

(st − st−∆

∆+ δ

),

where the increasing returns to scale nature of the cost function is now captured by the max term.12

During the lifetime of each generation, a polity with state strength st and civil society strength

xt produces output/surplus given by

f(xt, st), (3)

where f is assumed to be nondecreasing and differentiable.13 The dependence of the total output

of the economy on the strength of the state captures the various efficiency-enhancing roles of state

capacity. In addition, we allow for output to depend on the strength of civil society as well, which

might be because a strong civil society prevents extractive uses of the capacity of the state that

tend to reduce the total output or surplus in the economy, or because its greater cooperation and

coordination improves economic efficiency.

We next discuss how the output of society is distributed between the elite (controlling the state)

and citizens. At date t, if the elite and civil society (citizens) decide to fight, then one side will win

and capture all of the output of the economy, and the other side receives zero. Winning probabilities

are functions of relative strengths. In particular, the elite will win if

st ≥ xt + σt,

where σt is drawn from the distribution H independently of all past events. We denote the density

of the distribution function H by h. The existence of the random term σt captures the fact that

12Note that when we consider the limit ∆→ 0, we obtain

Cx(xt) = cx (xt + δ) + max {γx − xt, 0} (xt + δ),

Cs(st) = cs (st + δ) + max {γs − st, 0} (st + δ).

13The fact that (3) refers to output during the lifetime of each generation means that each generation will producethis quantity regardless of ∆ > 0. As we show more explicitly in footnote 14, this feature is important to ensure that theincentives for investment do not vanish when we consider short-lived players as in the next section and ∆→ 0. (Whenwe return to long-lived, forward-looking players, incentives for investment will not vanish and similar results apply as∆→ 0 even if (3) is multiplied with the period of length ∆; see footnote 13).

8

various stochastic factors impact the outcome of any conflict. Throughout, since both sides have the

same assessment of the outcome of conflict, we will presume that they divide total output according

to their expected shares, but whether they do so or actually engage in conflict is immaterial for our

results.

This specification of the stochastic contest function, and a symmetry assumption which we will

shortly impose, implies that when the strengths of civil society and state are given, respectively, by

x and s, the probability that the state will win the conflict is H(s− x), and the probability that the

civil society will do so is 1−H(s− x) = H(x− s), a property we will use frequently below.

In the Appendix, we also show that the most important qualitative features implied by this

formulations of conflict between the elite (state) and society are shared by other formulation of the

contest between these parties.

2.2 Assumptions

We next introduce three assumptions. The first one is a simplifying assumption, which we impose

initially and then relax subsequently:

Assumption 1 f(x, s) = 1 for all x ∈ [0, 1] and s ∈ [0, 1].

This assumption makes it transparent that the multiple steady-state equilibria and their dynamics

— our main focus — are driven by the dynamic contest between the state and civil society, not because

of changes in the value of the prize in this contest. It will be relaxed in Section 5.

The next two assumptions are imposed throughout.

Assumption 2 1. cx and cs are continuously differentiable, strictly increasing and weakly convex

over R+, and satisfy limx→∞ c′x(x) =∞ and limx→∞ c

′s(s) =∞.

2.

c′s(δ) 6= c′x(δ).

3.|c′′s(δ)− c′′x(δ)|

min{c′′x(δ), c′′s(δ)}<

1

supz |h′(z)|.

4.

c′s(0) + γs ≥ c′x(δ) and c′x(0) + γx > c′s(δ).

Part 1 of Assumption 2 is standard. Part 2 is imposed for simplicity and rules out the non-generic

case where the marginal cost of investment at δ is exactly equal for the two parties. Part 3 is also

imposed for technical convenience, and is quite weak. For example, if the gap between c′′x(δ) and c′′s(δ)

is small, this condition is automatically satisfied. We will flag its role when we come to our analysis,

but anticipating that discussion, it makes it much easier for us to establish the instability of some

“uninteresting” steady-state equilibria. Part 4 ensures that the marginal cost of each player in the

increasing returns region (when x < γx or s < γs) when making zero investment is greater than the

marginal cost of the other player outside this region when evaluated at δ — the marginal cost on the

right-hand side is evaluated at δ since, as our above transformation showed, the level of investment

9

necessary for maintaining any positive steady-state level of capacity is δ. We will flag the role of this

assumption when we come to our formal analysis.

Assumption 3 1. h exists everywhere, and is differentiable, single-peaked and symmetric around

zero.

2. For each z ∈ {x, s},c′z(0) > h(1).

3. For each z ∈ {x, s},min{h(0)− γz;h(γz)} > c′z(δ).

Part 1 contains the second key assumption for our analysis — single peakedness and symmetry

of h around 0 (differentiability is standard). This assumption not only simplifies our analysis as it

ensures that h(x − s) = h(s − x) and h′(x − s) = −h′(s − x), but also implies that incentives for

investment are strongest when x and s are close to each other. We highlight the role of this feature

below as well.

Part 2 imposes that when a player has the maximum gap between itself and the other player, it

has no further incentives to invest. Part 3, on the other hand, ensures that at or near the point where

conflict capacities are equal, there are sufficient incentives to increase conflict capacity. Both of these

assumptions restrict attention to the part of the parameter space of greater interest to us.

3 Equilibrium with Short-Lived Players

We now present our main results about the dynamics of the power of state and civil society, focusing

on the non-overlapping generations setup, where at each point in time, each side of the conflict is

represented by a single short-lived agent who will be replaced by a new agent from the same side next

period. This ensures that when players take decisions today they do not internalize their impact on

the future evolution of the power of either party.

3.1 Preliminaries

Suppose that the above-described society is populated by non-overlapping generations of agents —

on the one side representing the elite (state) and on the other, civil society.

With these assumptions, at each time t, civil society maximizes

H(xt − st)−∆ · Cx(xt, xt−∆)

by choosing xt (or equivalently ixt ), taking xt−∆ as given. Simultaneously, the elite maximize

H(st − xt)−∆ · Cs(st, st−∆)

by choosing st, taking st−∆ as given. A dynamic (Nash) equilibrium with short-lived players is given

by a sequence {x∗k∆, s∗k∆}

∞k=0 such that x∗k∆ is a best response to s∗k∆ given x∗(k−1)∆, and likewise s∗k∆

is a best response to x∗k∆ given s∗(k−1)∆.

10

The investment decisions of both elites and civil society are then determined by their respective

first-order conditions (with complementary slackness). In particular, at time t, we have:14

h(xt − st) ≤ c′x(xt−xt−∆

∆ + δ) + max{0; γx − xt−∆} ifxt−xt−∆

∆ = −δ or xt = 0,

h(xt − st) ≥ c′x(xt−xt−∆

∆ + δ) + max{0; γx − xt−∆} if xt = 1,

h(xt − st) = c′x(xt−xt−∆

∆ + δ) + max{0; γx − xt−∆} otherwise,

andh(st − xt) ≤ c′s(

st−st−∆

∆ + δ) + max{0; γs − st−∆} ifst−st−∆

∆ = −δ or st = 0,

h(st − xt) ≥ c′s(st−st−∆

∆ + δ) + max{0; γs − st−∆} if st = 1,

h(st − xt) = c′s(st−st−∆

∆ + δ) + max{0; γs − st−∆} otherwise.

The first line of either expression applies when the relevant player has chosen zero investment so that

its state variable shrinks as fast as it can (at the rate δ), or is already at its lower bound xt = 0 or

st = 0. In this case, we have the additional cost of investment on the right-hand side, and also the

optimality condition is given by a weak inequality, since at this lower boundary, the marginal benefit

could be strictly less than the marginal cost of investment. The second line, on the other hand, applies

when the state variable takes its maximum value, 1, and in this case the marginal benefit could be

strictly greater than the marginal cost of investment. Away from these boundaries, the third line

applies and requires that the marginal benefit equal the marginal cost. Note also that the marginal

benefit for civil society is the same as the marginal benefit for the state — since h(st−xt) = h(xt−st).On the other hand, we also have from Assumption 3 that changes in the marginal benefits of the two

players are the converses of each other — that is, h′(st − xt) = −h′(xt − st).Now letting ∆ → 0, we obtain the following continuous-time first-order optimality (and thus

equilibrium) conditions

h(xt − st) ≤ c′x(xt + δ) + max{0; γx − xt} if xt = −δ or xt = 0,h(xt − st) ≥ c′x(xt + δ) + max{0; γx − xt} if xt = 1,h(xt − st) = c′x(xt + δ) + max{0; γx − xt} otherwise,

(4)

andh(st − xt) ≤ c′s(st + δ) + max{0; γs − st} if st = −δ or st = 0,h(st − xt) ≥ c′s(st + δ) + max{0; γs − st} if st = 1,h(st − xt) = c′s(st + δ) + max{0; γs − st} otherwise.

(5)

In what follows, we work directly with these continuous-time first-order optimality conditions. More-

over, it is straightforward to see that in continuous time, away from the boundaries of [0, 1]2 these

first-order optimality conditions will hold as equality, and thus the dynamics of state and civil society

strength can be represented by the following two differential equations:

x = max{(c′x)−1(h(x− s)−max{γx − x, 0}); 0} − δ (6)

s = max{(c′s)−1(h(s− x)−max{γs − s, 0}); 0} − δ.

The roles of the two key assumptions highlighted above — the single-peakedness of h and the

increasing returns aspect of the cost function — are evident from (6). First, when x and s are close

14Following up on footnote 13, we can more clearly see the role that ∆ in front of the cost function plays here:without this term (or equivalently if the return was also multiplied by ∆), the marginal cost of investment would bemultiplied by 1/∆, and thus as ∆→ 0, investments would converge to zero. This is because short-lived players that arenot forward-looking do not take the impact of their instantaneous investments on future stocks (and have infinitesimalimpact on the current stock).

11

to each other, h(x−s) is large, and thus both of these variables will tend to grow further. Conversely,

when x and s are far apart, h(x − s) is small, and investment by both parties is discouraged. This

observation captures the key economic force that will lead to the emergence of different dynamics of

state-society relations and different types of states in our setup (in the Appendix, we see that this

same property holds with other formulations of the contest function). Secondly, the presence of the

max term implies that once the conflict capacity of a party falls below a critical threshold (γx or γs),

there is an additional force pushing towards further reduction in this capacity.

3.2 Dynamics of the Strength of Civil Society and the State

Our main result in this section is summarized in the next proposition.

Proposition 1 Suppose Assumptions 1, 2 and 3 hold. Then there are three (locally) asymptotically

stable steady states:

1. x∗ = s∗ = 1.

2. x∗ = 0 and s∗ ∈ (γs, 1).

3. x∗ ∈ (γx, 1) and s∗ = 0.

This proposition shows that there exist three relevant (asymptotically stable) steady states, one

corresponding to an inclusive state, one corresponding to a despotic state and one to a weak state.

The intuition, as already anticipated, is that when we are in the neighborhood of the steady state

x∗ = s∗ = 1, h(x − s) is large, encouraging both parties to move further towards x∗ = s∗ = 1. In

contrast, in the neighborhood of x∗ = 0 or s∗ = 0, h(x− s) is small, and neither party has as strong

incentives to invest, and in fact, one of them ends up with zero conflict capacity.15

The steady states presented in Proposition 1 and their local dynamics are depicted in Figure

2. Our analysis so far establishes the dynamics in the neighborhoods of these steady states. After

providing the proof of Proposition 1, we turn to a characterization of global dynamics.

3.3 Proof of Proposition 1

We start with a series of lemmas on the steady-state equilibria of this model, and their stability

properties. Before presenting these results, we remark that, mathematically, there can be three types

of steady states: (i) those in which the party in question (say the civil society) chooses a positive

level conflict capacity, and thus we will have x∗t = x∗ ∈ (0, 1), so that the marginal cost of investment

is simply c′x(δ) + max{γx − x∗, 0}, which is equal to the benefit from this conflict capacity; (ii) those

in which we have zero conflict capacity, in which case the marginal cost of investment, c′x(0) + γx, is

greater than or equal to the benefit from building further conflict capacity; (iii) those in which the

party in question has conflict capacity equal to 1, in which case marginal cost of investment, c′x(δ), is

less than or equal to the benefit from building additional conflict capacity.

15Under Assumption 1, there is no social benefit in reaching the steady state x∗ = s∗ = 1, since the capacities of thestate and society do not contribute to the size of the social surplus. This will be relaxed below when we consider thegeneral environment in which x and s contribute to the size of total surplus.

12

Power of Society

Power of the State

0 1x∗0

1

s∗

Region I

Region II

Region III

Figure 2: Steady states and their local dynamics.

Lemma 1 There exists a (locally) asymptotically stable steady state with x∗ = s∗ = 1.

Proof. At x∗ = s∗ = 1, the marginal cost of investment for player z ∈ {x, s} is c′z(δ), while the

marginal benefit starting from this point is h(0), so Assumption 3 ensures that the marginal benefit

strictly exceeds the marginal cost, and neither player has an incentive to reduce its investment.

Furthermore, because 1 is the maximum level of investment, neither party has the ability to increase

it.

We turn next to asymptotic stability of this steady state. First note that from (6), the laws of

motion of x and s in the neighborhood of the state state (x∗ = 1, s∗ = 1) are given by

c′x(x+ δ) = h(x− s) if x < 1 and x = 0 if x = 1 (7)

c′s(s+ δ) = h(s− x) if s < 1 and s = 0 if s = 1,

where we are exploiting the fact that once we are away from the steady state, there cannot be an

immediate jump and thus the first-order conditions have to hold in view of Assumption 2. We have

also used the information that we are in the neighborhood of the steady state (1, 1) in writing the

system for x > γx and s > γs. Now to establish asymptotic stability, we will show that

L(x, s) =1

2(1− x)2 +

1

2(1− s)2

is a Lyapunov function in the neighborhood of the steady state (1, 1). Indeed, L(x, s) is continuous

and differentiable, and has a unique minimum at (1, 1). We next verify that in a sufficiently small

13

neighborhood of (1, 1), L(x, s) is decreasing along solution trajectories of the dynamical system given

by (7). Since L is differentiable, for x ∈ (γx, 1) and s ∈ (γs, 1), we can write

dL(x, s)

dt= −(1− x)x− (1− s)s.

First note that since h(x − s) > c′x(δ) and h(s − x) > c′s(δ) for x and s in a sufficiently small

neighborhood of (1, 1), we have both x > 0 and s > 0. This implies that, in this range, both

terms in dL(x,s)dt are negative, and thus dL(x,s)

dt < 0. Moreover, the same conclusion applies when

x = 1 (respectively when s = 1), with the only modification that dL(x,s)dt no longer includes the s

(respectively the x) term, but still continues to be strictly negative, even on the boundary of [0, 1]2.

Then the asymptotic stability of (1, 1) follows from LaSalle’s Theorem (which takes care of the fact

that our steady state is on the boundary of the domain of the dynamical system in question, see, e.g.,

Walter, 1998).

This lemma shows that under our maintained assumptions, both parties investing at their maxi-

mum conflict capacity is a steady-state equilibrium. Intuitively, this proposition exploits the fact that

when the two players are “neck and neck,” they both have strong incentives to invest. If instead we

had, say, x much larger than s, then from part 1 of Assumption 3, both h(x− s) and h(s− x) would

be smaller than h(0), reducing the investment incentives of both parties. The stronger investment

incentives around x∗ = s∗ = 1 are key for maintaining this combination as an (asymptotically stable)

steady state — combined with part 2 of Assumption 3, which ensures that these strong incentives

are sufficient to guarantee a corner solution. If the inequality in part 2 of Assumption 3 did not hold,

x∗ = s∗ = 1 could not be a steady-state equilibrium, and in this case, the only possible steady-state

equilibria would be those identified in Lemma 2 below.

The local stability of this steady state is then established by constructing a Lyapunov function.

The use of this method is necessitated by the fact that x∗ = s∗ = 1 is at the corner of the feasible

set, [0, 1]2, and thus dynamics around it cannot be characterized by using linearization methods.

Our next result identifies two additional locally asymptotically stable steady states.

Lemma 2 There exist two additional (locally) asymptotically stable steady states:

1. one with x∗ = 0 and s∗ ∈ (γs, 1), and

2. one with s∗ = 0 and x∗ ∈ (γx, 1).

Proof. We start with the first statement. Suppose first that x∗ = 0. Then from (5) an interior

steady-state level of investment for the state requires

h(s) = c′s(δ) + max{0; γs − s}.

Note that Assumption 3 implies that at s = 1, h(1) < c′s(δ), and at s = γs, h(γs) > c′s(δ), thus by

the intermediate value theorem, there exists s∗ between γs and 1 satisfying

h(s∗) = c′s(δ). (8)

Moreover, because h is single peaked and symmetric around 0, h(s) is decreasing in s ≥ γs, and thus

only a unique s∗ satisfying this relationship exists.

14

We next verify that x∗ = 0 is indeed consistent with the optimization of civil society. This follows

immediately since

h(−s∗) = h(s∗) = c′s(δ) < c′x(0) + γx,

where the first equality follows from the symmetry of h, the second one simply replicates (8), and the

inequality follows from Assumption 2, and establishes that x∗ = 0 is optimal for civil society.

The local stability is again established using a Lyapunov argument as in the proof of Lemma 1.

Now in the neighborhood of the state state (x = 0, s = s∗), the dynamical system in (6) can be

written as

c′x(x+ δ) = h(x− s) + γx − x if x > 0 and x = 0 if x = 0, and

c′s(s+ δ) = h(s− x),

where we are now using the fact that we are in the neighborhood of (0, s∗) so that x < γx and s > γs.

The dynamical system in (6) in this case can be written as

x = (c′x)−1(h(x− s) + γx − x)− δ (9)

s = (c′s)−1(h(s− x))− δ.

We now choose the Lyapunov function

L(x, s) =1

2x2 +

1

2(s− s∗)2,

which is again continuous and differentiable, and has a unique minimum at (0, s∗). We next verify

that in the neighborhood of (0, s∗), L(x, s) is decreasing along solution trajectories of the dynamical

system given by (9). Specifically, since L is differentiable, for x ∈ (0, γx) and s ∈ (γs, 1), we can write

dL(x, s)

dt= xx+ (s− s∗)s.

First note that as h(−s∗) < c′x(δ) + γx, for x and s in the neighborhood of (0, s∗),

x = (c′x)−1(h(x− s) + γx − x)− δ < 0. (10)

Then, using a first-order Taylor expansion of (9) in this neighborhood, we obtain

(s− s∗)s =1

c′′s(δ)h′(s∗)(s− s∗)(s− x− s∗) + o(·), (11)

where o(·) denotes second-order terms in x and s− s∗.The desired result follows from the following arguments: (i) for x ∈ (0, γx) and s ∈ (γs, 1), |xx| >

|(s− s∗)s|, regardless of the sign of (s−s∗)s, as x→ 0 and s→ 0, (s−s∗)(s−x−s∗)/x→ 0, because

in the neighborhood of the steady state (0, s∗), s is of the order s − s∗, while h(−s∗) < c′x(δ) + γx,

ensuring that x < 0). Therefore, in the range where x ∈ (0, γx) and s ∈ (0, γs),dL(x,s)dt < 0. (ii) when

x = 0, (11) implies that (s−s∗)s < 0 in view of the fact that h′(s∗) < 0, and thus we have dL(x,s)dt < 0.

(iii) when s = s∗, (10) ensures that x < 0, so that we again have dL(x,s)dt < 0. Then in all three cases,

the asymptotic stability of (0, s∗) follows from LaSalle’s Theorem (e.g., Walter, 1998).

15

The proof of the existence, uniqueness and asymptotic stability of the steady state with s∗ = 0

and x∗ ∈ (γx, 1) is analogous, and is omitted.

These two additional steady states have a very different flavor than the steady state in Lemma

1. Now both parties have a lower level of conflict capacity, and one of them is in fact at zero. The

intuition is again related to the incentives for investment in conflict capacity: when one party is at

zero capacity, h(·) is small for both players, which encourages the first player to build a state with

low capacity, and discourages the other player from building further capacity.

Assumptions 2 and 3 play an important role in this lemma as well. Without the boundary condi-

tions in Assumption 3, there could be other steady states with some of them including investments

below γx and γs. Though these steady states would be locally unstable (with the same argument as

in Lemma 4 below), it would also become harder to ensure that there exists a locally stable steady

state, making us prefer these assumptions.

The next lemma rules out several types of steady states.

Lemma 3 There is no steady state with (i) x∗ = s∗ = 0; or (ii) x∗ = 0 and s∗ ∈ (0, γs), or s∗ = 0

and x∗ ∈ (0, γx); or (iii) x∗ ∈ (γx, 1) and s∗ ∈ (γs, 1).

Proof. Claim (i) follows immediately, since from part 3 of Assumption 3, we have h(0)−γs > c′s(0),

so that when x∗ = 0, the elite will deviate from s = 0. Claim (ii) follows directly from the proof of

Lemma 2. Finally, for claim (iii), note that a steady state with x∗ ∈ (γx, 1) and s∗ ∈ (γs, 1) would

necessitate

h(s∗ − x∗) = c′s(δ) (12)

h(x∗ − s∗) = c′x(δ),

but then from the symmetry of the h function around zero, we have that h(s∗ − x∗) = h(x∗ − s∗), so

that

c′s(δ) = h(s∗ − x∗) = c′x(δ),

which contradicts part 2 of Assumption 2.

There are other types of steady states that could exist, but the next lemma shows that when they

do, they will all be asymptotically unstable.

Lemma 4 All other (possible) steady states are asymptotically unstable.

Proof. We will prove this lemma by considering three types of steady states, which exhaust all

possibilities.

Type 1: x∗ ∈ (0, γx) and s∗ ∈ (0, γs).

The optimality conditions in such a steady state are

h(s∗ − x∗) = c′s(δ) + γs − s∗

h(x∗ − s∗) = c′x(δ) + γx − x∗.

16

The dynamical system (6) now becomes

x = (c′x)−1(h(x∗ − s∗) + γx − x∗)− δ

s = (c′s)−1(h(s∗ − x∗) + γs − s∗)− δ.

Since the steady-state levels of state and civil society strength are defined by equality conditions in

this case, local dynamics can be determined from the linearized system, with characteristic matrix

given by (1

c′′s (δ) [h′(s∗ − x∗) + 1] − 1c′′s (δ)h

′(s∗ − x∗)− 1c′′x(δ) [h′(x∗ − s∗) 1

c′′x(δ) [h′(x∗ − s∗) + 1]

).

Using the fact that from Assumption 3, h′(s∗ − x∗) = −h′(x∗ − s∗), the determinant of this matrix

can be computed as 1c′′s (δ)c′′x(δ) > 0. Moreover, from part 2 of Assumption 2, we can show that the

trace of this matrix is1

c′′s(δ)[h′(s∗ − x∗) + 1] +

1

c′′x(δ)[h′(x∗ − s∗) + 1].

Once again using Assumption 3, this expression is positive provided that

h′(s∗ − x∗)(c′′s(δ)− c′′x(δ)) ≤ c′′x(δ) + c′′s(δ). (13)

Assumption 2 ensures that ∣∣c′′s(δ)− c′′x(δ)∣∣ ≤ c′′x(δ)

|h′(s∗ − x∗)|,

which is a sufficient condition for (13), establishing that both eigenvalues are positive, and we have

asymptotic instability.

Type 2: x∗ ∈ (γx, 1) and s∗ ∈ (0, γs), or x∗ ∈ (0, γx) and s∗ ∈ (γs, 1). Consider the first of these,

h(s∗ − x∗) = c′s(δ) + γs − s∗

h(x∗ − s∗) = c′x(δ).

Now once again, local dynamics can be determined from the linearized system, with characteristic

matrix (1

c′′s (δ) [h′(s∗ − x∗) + 1] − 1c′′s (δ)h

′(s∗ − x∗)− 1c′′x(δ) [h′(x∗ − s∗) 1

c′′x(δ)h′(x∗ − s∗)

).

The trace of this matrix is

1

c′′s(δ)[h′(s∗ − x∗) + 1] +

1

c′′x(δ)h′(x∗ − s∗),

which is positive provided that

h′(s∗ − x∗)(c′′s(δ)− c′′x(δ)) ≤ c′′x(δ).

The same argument as in the proof of Type 1 shows that this condition follows from Assumption 2,

implying that at least one of the eigenvalues is positive and thus establishing asymptotic instability.

The argument for the case where x∗ ∈ (0, γx) and s∗ ∈ (γs, 1) is analogous.

Type 3: s∗ = 1 and x∗ < 1 or x∗ = 1 and s∗ < 1.

17

We prove the first case (the proof for the second is analogous). Such a steady state exists only if

h(1− x∗) ≥ c′s(δ)

h(x∗ − 1) = c′x(δ) + max{γx − x∗, 0}.

Exploiting these conditions, we will show that such a steady state cannot be asymptotically stable.

To do this, let us distinguish between x∗ > γx and x∗ ≤ γx. Consider the first one of these. Then

consider a perturbation that keeps s∗ constant and reduces x∗ to x∗ − εx for εx > 0 small (since it is

sufficient to show asymptotic instability for a specific set of perturbations). Then, we have

x = − 1

c′′x(δ)h′(x∗ − 1)− δ < 0.

The sign follows because h′(x∗ − 1) > 0 from Assumption 3, and implies that x∗ decreases away

from the steady state in question, establishing asymptotic instability. Consider finally the second

possibility, with the same perturbation which yields

x = − 1

c′′x(δ)[h′(x∗ − 1) + 1]− δ < 0,

which is also locally asymptotically unstable. This completes the proof of the lemma.

Proposition 1 then follows straightforwardly by combining these lemmas.

3.4 Global Dynamics

We next partially characterize the global dynamics. In particular, we will determine three regions, as

shown in Figure 3, separating the phase diagram into basins of attraction of the three asymptotically

stable steady states characterized in the previous subsection. For example, starting from Region I,

equilibrium dynamics converge to the steady state with x∗ = 0 and s∗ ∈ (γs, 1); from Region II,

convergence is to the steady state with x∗ = s∗ = 1; and from Region III, convergence will be to

the steady state with x∗ ∈ (γx, 1) and s∗ = 0. Unfortunately, it is not possible to determine the

boundaries of these regions analytically, but we will be able to characterize subsets thereof explicitly.

Consider first Region II, which is the basin of attraction of the steady state x∗ = s∗ = 1. Recall

that the dynamical system for the behavior of the conflict capacity of civil society and state take the

form given in (6) above. We proceed by first noting that any subset S of [0, 1]2 for which there exists

a Lyapunov function L(x, s) such that (i) S = {(x, s) : L(x, s) ≤ K} for some K > 0; (ii) L(x, s) ≥ 0

for all (x, s) ∈ S, with equality only if x = s = 1; and (iii) ∂L(x, s)/∂t ≤ 0 for all (x, s) ∈ S, with

equality only if x = s = 1, is part of the basin of attraction of this steady state.

Let us first construct a subset of the parameters (x, s) such that x ≥ 0 and s ≥ 0, with one

of them holding as strict inequality. Let us define x such that c′x(δ) = h(x − 1). Clearly, from

Assumption 2 c′s(δ) < h(1 − x). This defines R′′II= {(x, s) : x ≥ max{γx, x} and s ≥ max{γs, x}}.This region can be further extended by noting that any combination of (x, s) such that (c′x)−1(h(x−s)−max{γx − x, 0})− δ ≥ 0 and (c′s)

−1(h(s− x)−max{γs − s, 0})− δ ≥ 0 also satisfies x ≥ 0 and

s ≥ 0. Let us define s(x) such that h(s(x) − x) − max{γs − s(x), 0} − c′s(δ) = 0. Similarly, define

x(s) such that h(x(s)− s)−max{γx− x, 0}− c′x(δ) = 0. Both s(x) and x(s) are upward sloping, and

in fact correspond to lines with slope 1 when s ≥ γs and x ≥ γx, respectively. Then starting within

18

Power of Society

Power of the State

0 1x∗

x(s)

0

1

s∗s(x)

Figure 3: Global dynamics.

R′II = {(x, s) : s ≤ s(x) and x ≤ x(s)}, we also have x ≥ 0 and s ≥ 0 (and in fact, R′′II ⊂ R′II). This

region, as well as R′′II , is depicted in Figure 3. The shape of the region is intuitive.

Now consider the family of functions, L(x, s | lx, ls) = lx2 (1− x)2 + ls

2 (1− s)2, indexed by lx > 0

and ls > 0. Clearly, for any member of this family, we have that for all (x, s) ∈ R′II\(1, 1),

∂L(x, s | lx, ls)∂t

= lx(1− x)x− ls(1− s)s < 0.

So if we in addition define the subset RII of R′II where L(x, s | lx, ls) ≤ K, then RII satisfies the

above conditions and by construction is part of the basin of attraction of the steady state (1, 1).

Now consider the problem of choosing K, lx and ls such that we achieve the largest set RII =

{(x, s) : L(x, s | lx, ls) ≤ K} contained in R′II . Mathematically, let A(RII) be the area of set RII .Then the problem is to choose

maxK,lx,ls>0

A(RII).

Figure 3 shows the construction of region RII in this manner, which is by construction part of the

basin of attraction of the steady state (1, 1).

Subsets of the basins of attraction of the other steady states can be constructed analogously and

are shown in Figure 3.

We also verify numerically that dynamics take the form shown in Figures 2 and 3. In Figure 4,

we depict the vector field for a specific parameterization of the model. We take f(x, s) = 0.6, and

choose H to be a raised cosine distribution over [−1, 1] with mean µ = 0, which is single-peaked and

19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

Figure 4: The direction of change of the power of state and society in a simulated example.

symmetric consistent with Assumption 3.16 The cost functions of the state and civil society are

cx(i) = 3.25× i2 (for i ∈ [0, 10]) and cs(i) = 3.25× i2 (for i ∈ [0, 15]),

and outside of these ranges, the cost functions become vertical, placing a bound on investment

levels.17 In addition, we set γx = 0.35, γs = 0.35, and δ = 0.1. The figure verifies the qualitative

characterization provided so far.

3.5 The “Conditional” Effects of Changes in Initial Conditions

Though we will discuss comparative statics (or “comparative dynamics”) in greater detail in Section

5, here we undertake a simple exercise: change the initial conditions and trace the effects of these on

equilibrium dynamics. The immediate but important conclusion is that the same change in initial

conditions, starting from different parts of the state space, can have drastically different implications.

Consider an increase in s0 to s0 + s. This can leave us in the same region as before, in which

case the equilibrium trajectory will be shifted uniformly up, but the long-run outcome will remain

unchanged. Alternatively, this increase can shift us from, say, Region III to Region II, in which

case not only the equilibrium trajectory but also the long-run outcome will change, and in fact it

will involve greater state capacity. However, depending on the exact value of (x0, s0), an increase of

the same amount s could also shift us from Region II to Region I, in which case the impact on the

16Assumption 1 imposed that f(x, s) = 1 rather than setting it equal to a constant, say φ0, in order to reduce thenumber of parameters. We consider a more general surplus function in Assumption 1′ below. Setting φ0 = 0.6 enablesus to construct an example with more equally-sized regions.

17This bound plays no role in the numerical results reported here, but facilitates convergence when we consider thedynamic model with the same parameterization and low discount rates in the next section.

20

long-run state capacity will be negative instead of positive. This illustrates, while also providing a

simple proof, that the effects of changes in initial conditions in this model are conditional — they

depend exactly on where we start. This discussion thus establishes:

Proposition 2 The effects of changes in the initial conditions (x0, s0) on equilibrium dynamics and

the long-run outcome of the society are conditional in the sense that these depend on which region we

move out of and into.

4 Equilibrium with Forward-Looking Players

In this section, we analyze our general framework with long-lived, forward-looking players. After

briefly describing preferences, we first show that for high rates of discounting, equilibrium behavior

converges to behavior with short-lived players, which we characterized in the previous section. We

then numerically study the dynamics of the equilibrium for a range of discount rates. In this section,

we continue to impose Assumption 1, and then relax it in the next section.

4.1 Preferences

We start with the discrete time model. The technology of investment and conflict are the same as in

the previous two sections. The only difference is that now both civil society and state are long-lived

and forward-looking. To maximize the parallel with the model with short-lived players, we assume

that both players again correspond to sequences of non-overlapping generations, but each generation

has an exponentially-distributed lifetime or equivalently, a Poisson end date with parameter, 1 − β,

where β = e−ρ∆. We assume that this random end date is the only source of discounting. Clearly,

this specification guarantees that as the period length ∆ shrinks, discounting between periods will

also decline (and the discount factor will approach 1). Again to maximize the parallel with our

static model, we assume that in expectation, there is one instance of conflict between the two players

during the lifetime of each generation. Since with this Poisson specification, the expected lifetime of

his generation is 1/(1− β), this implies that a conflict arrives at the rate 1− β.18

4.2 Main Result

The main result we prove in this forward-looking model is provided in the next proposition.

Proposition 3 Suppose Assumptions 1, 2 and 3 hold. Then there exist discount rates ρ ≥ ρ > 0

such that for all ρ > ρ, there are three (locally) asymptotically stable steady states:

1. x∗ = s∗ = 1.

2. x∗ = 0 and s∗ ∈ (γs, 1).

18An alternative specification of the model with long-lived players which leads to identical equations, but eschews theparallel with the static model, is to assume that both players are infinitely lived and discount the future at the rateβ = e−ρ∆ and there is a conflict during each interval of length ∆. Recall from footnote 14 that in this case there will beno investment when ∆→ 0 with short-lived players (because they do not take into account the benefit from increasingfuture conflict capabilities), but incentives for investment do not disappear with long-lived, forward-looking players evenas ∆→ 0 (because they do take into account the benefit from increasing future conflict capabilities).

21

3. x∗ ∈ (γx, 1) and s∗ = 0.

Moreover, for all ρ < ρ, there exists a unique globally stable steady state x∗ = s∗ = 1.

This result thus shows that the main insights from our analysis apply provided that players, though

forward-looking, are sufficiently impatient. We note that this result is not a simple consequence of

the fact that as we consider larger and larger values of ρ, players are becoming closer to myopic. It

necessitates establishing properties of the the relevant value functions and their derivatives in the

limit.19

4.3 Proof of Proposition 3

With the specification introduced above, we can straightforwardly represent the maximization prob-

lem of each player as a solution to a recursive, dynamic programming problem, written as

Vx(xt−∆, st−∆, β; ∆) = maxxt∈[0,1]

{(1− β)H(xt − st)−∆ · Cx (xt, xt−∆) + βVx(xt, s

′∗(xt−∆, st−∆, β; ∆), β; ∆)},

(14)

and

Vs(xt−∆, st−∆, β; ∆) = maxst∈[0,1]

{(1− β)H(st − xt)−∆ · Cs (st, st−∆) + βVs(x

′∗(xt−∆, st−∆, β; ∆), st, β; ∆)}.

(15)

Several things are important to note. First, as anticipated in the previous section, we multiply the

flow costs with ∆, but not the benefits, since these capture life-time benefits from conflict, and we

have conditioned on ∆ in writing the value functions for emphasis. Second, notice that we have

already imposed the boundary conditions, xt ∈ [0, 1] and st ∈ [0, 1], in the maximization problems.

Third, x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆) are the policy functions, which give the next period’s values

of the state variables as a function of this period’s values (and are explicitly conditioned on ∆ > 0).

A dynamic equilibrium in this setup as given by a pair of policy functions, x′∗(x, s, β; ∆) and

s′∗(x, s, β; ∆) which give the next period’s values of the state variables as a function of this period’s

values (for ∆ > 0), and each solves the corresponding value function taking the policy function of the

other party is given. Once these policy functions are determined, the dynamics of civil society and

state strength can be obtained by iterating over these functions.

Since these are standard Bellman equations, the following result is immediate (throughout this

proof we take (x, s, β) ∈ [0, 1]3).

Lemma 5 For any ∆ > 0, Vx(x, s, β; ∆) and Vs(x, s, β; ∆) exist and are continuously differentiable

in x, s and ∆. Moreover, x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆) and are continuous in x, s and ∆.

In particular, from (14) and (15), as β → 0, Vx(x, s, β; ∆)→ Vx(x, s, β = 0; ∆) and Vs(x, s, β; ∆)→Vs(x, s, β = 0; ∆). But since x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆) are maximizers of the continuous

(and bounded) functions, (14) and (15), we can apply Berge’s maximum theorem to conclude that

x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆) are also continuous, particularly in β, and thus x′∗(x, s, β; ∆) →19When ρ is between ρ and ρ, we may have a situation in which one of the two corner steady states disappears while

the other one still exists.

22

x′∗(x, s, β = 0; ∆) and s′∗(x, s, β; ∆) → s′∗(x, s, β = 0; ∆), and thus for β sufficiently close to 0, we

have that x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆) are approximately the same as their myopic values. There-

fore, there exists β > 0, such that for all β < β, a steady state of the dynamical system given by

x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆) exists and is locally stable if and only if it is a locally stable steady

state of the myopic model.

This argument establishes that the forward-looking, discrete-time dynamics when the discount

factor is sufficiently close to 0 will have the same locally stable steady states as the myopic, discrete-

time dynamics. In the previous section, we approximated the discrete-time dynamics with their

continuous-time limit, and it is also convenient to do the same here, and to maximize the parallel,

this is how we have stated the proposition.

We can also observe that when the discount factor β → 1, the two steady states other than (1, 1)

disappear. The argument is simple: take the steady state with x = 0, where the society’s flow return

is zero. If civil society invests at a high level for a finite number of periods, this will ensure that

x ≥ γx, eliminating the region of higher costs of investment for civil society, and thus taking x to 1

(which gives the society a positive flow return). When β is arbitrarily close to 1, the costs of investing

at a high level for a finite number of periods are negligible, and hence such a deviation is profitable

for civil society. This argument, again from continuity, ensures that there exists βx < 1 such that

for β > βx, x = 0 is not consistent with a steady state. With the parallel argument, we also have

that there exists βs < 1 such that for β > βs, s = 0 cannot be part of a steady state. Then, for

β > β = max{βx, βs} only (1, 1) remains as an asymptotically stable steady state.

The next subsection discusses the continuous-time limit and also derives the continuous-time

Hamilton-Jacobi-Bellman (HJB) equations, which can be used to characterize the equilibrium more

generally. We then come back to completing the proof of Proposition 3.

4.4 Continuous-Time Approximation

For characterizing the equilibrium for any value of the players’ impatience, we can once again use

the continuous-time approximation by taking the limit ∆→ 0, which shrinks the period length (and

correspondingly adjusts the discount factor β = e−ρ∆, so that the discount rate remains constant at

ρ). In this limit, conditions on β translate into conditions on ρ. More specifically, we have:

Lemma 6 As ∆ → 0, the value functions Vx(x, s, β; ∆) and Vs(x, s, β; ∆) converge to their

continuous-time limits Vx(x, s) and Vs(x, s), and the policy functions x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆)

converge to their continuous-time limits x′∗(x, s) and s′∗(x, s).20

Proof. This follows given the continuous differentiability of Vx(x, s, β; ∆) and Vs(x, s, β; ∆) and

of x′∗(x, s, β; ∆) and s′∗(x, s, β; ∆) for all ∆ > 0.

The continuous-time Hamilton-Jacobi-Bellman (HJB) equations can be obtained as follows. First

20We also drop the conditioning on the discrete-time discount factor β in writing the continuous-time value and policyfunctions and do not add conditioning on its continuous-time equivalent, the discount rate ρ to simplify the notation.

23

rearrange (14) evaluated at the optimal choices and divide both sides by ∆ to obtain

1− β∆

Vx(xt−∆, st−∆, β; ∆)

= maxxt≥0

[1− β

∆H(xt − st)− Cx (xt, xt−∆) + β

Vx(xt, s∗∆(xt−∆, st−∆), β; ∆)− Vx(xt−∆, st−∆, β; ∆)

∆

].

Now note that as ∆ → 0, (1 − β) → 0 and (1 − β)/∆ → ρ. Moreover the last term in the previous

expression tends to the total derivative of the value function with respect to time. Therefore, the

continuous-time HJB equation for civil society is

ρVx(x, s) = ρH(x− s) + maxx≥−δ

{−Cx(x, x) +

∂Vx(x, s)

∂xx

}+∂Vx(x, s)

∂ss∗(x, s),

where we have used the notation Cx(x, x) to denote the continuous-time cost function as a function of

the change in the conflict capacity of civil society, while x∗(x, s) and s∗(x, s) designate the continuous-

time policy functions, conveniently written in terms of the time derivative of the conflict capacities

of the two parties. We have also imposed that x cannot be less than −δ.Applying the same argument to (15) and denoting the continues-time cost function for the state

by Cs(s, s), we also obtain

ρVs(x, s) = ρH(s− x) + maxs≥−δ

{−Cs(s, s) +

∂Vs(x, s)

∂ss

}+∂Vs(x, s)

∂xx∗(x, s),

The first-order optimality conditions for civil society are given by

∂Cx(x, x)

∂x=∂Vx(x, s)

∂xif − δ < x(x, s), and x ∈ (0, 1),

∂Cx(x, x)

∂x≤ ∂Vx(x, s)

∂xif x = 1, (16)

∂Cx(x, x)

∂x≥ ∂Vx(x, s)

∂xif x(x, s) = −δ or x = 0.

In the first case, when we have an interior solution, we can also write

x =

(c′x)−1(∂Vx(x,s)

∂x − γx + x)

if x ≤ γx(c′x)−1

(∂Vx(x,s)

∂x

)if x > γx

. (17)

The first-order conditions for the state are also similar, and for an interior solution, they yield

s =

(c′s)−1(∂Vs(x,s)

∂s − γs + s)

if s ≤ γs(c′s)

−1(∂Vs(x,s)

∂s

)if s > γs

. (18)

4.5 Completing the Proof of Proposition 3

We have already established that, for fixed ∆ > 0, as β → 0, the solution and the implied dynamics

in the forward-looking case converge to their equivalents. This ensures that there exists β > 0, such

that for β < β, the dynamics of the myopic and forward-looking cases will be sufficiently close to

each other and thus will have the same set of locally stable steady states. Then taking the limit

∆→ 0, this can be expressed in terms of the continuous-time approximations for both systems, thus

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

Figure 5: The direction of change of the power of state and society in a simulated example withρ = 30.

establishing that for ρ > ρ, the locally stable steady states in Propositions 1 and 3 coincide. Similarly,

we also argue that for β > β (where β < 1), there cannot exist a steady state with either x = 0 or

s = 0, and thus the only asymptotically stable steady state is (1, 1). With the same argument, when

∆ → 0, we can then conclude that there exists ρ > 0 such that for discount rates below this, only

(1, 1) remains as the asymptotically stable steady state.

4.6 Numerical Results

We next provide a numerical characterization of the dynamics in the forward-looking model. We use

the same formulation of the cost function and parameter values as above. The critical threshold for

ρ is computed as ρ = 100, and for discount rates above this value, the vector field is identical to that

shown in Figure 4, confirming that for high discount rates the equilibrium dynamics of the model with

forward-looking agents coincide with the equilibrium of the model with myopic agents as claimed in

Proposition 1. In Figure 5, we also show the implied vector field when ρ is smaller than ρ (in this

instance, ρ = 30), which illustrates the very different dynamics in this case.

5 General Characterization

In this section, we relax Assumption 1. Since we have established the equivalence of the myopic and

forward-looking models when the discount rate is sufficiently large in the latter (which is a result

that does not depend in any way on Assumption 1), here we focus on a model with forward-looking

players. We also simplify the analysis throughout by assuming that f is linear as specified in the next

assumption, which replaces Assumption 1.

25

5.1 Modified Assumptions

Assumption 1′ f(x, s) = φ0 + φxx+ φss, where φ0 > 0, φx > 0 and φs > 0.

Our other two assumptions also require some minor modifications, which are provided next.

Assumption 2′ 1. cx and cs are continuously differentiable, strictly increasing and weakly convex,

and satisfy limx→∞ c′x(x) =∞ and limx→∞ c

′s(s) =∞.

2.

c′s(δ) 6= c′x(δ).

3.|c′′s(δ)− c′′x(δ)|

min{c′′x(δ), c′′s(δ)}< inf

z

2h(z)(φs + φx)

|h′(z)| (φ0 + φs + φx).

4.

c′s(0) + γs ≥ c′x(δ) and c′x(0) + γx > c′s(δ).

The minor modifications in parts 3 and 4 are in view of the fact that marginal benefits of investment

are different between the state and civil society. For same reason, we also modify Assumption 3 as

follows.

Assumption 3′ 1. h exists everywhere, and is differentiable, single-peaked and symmetric around

zero.

2. For each z ∈ {x, s},c′z(0) > h(1)(φ0 + φz) +H(1)φz.

3. For each z ∈ {x, s},

min{h(0)φ0 +H(0)φz − γz;h(γz)(φ0 + φzγz) +H(γz)φz} > c′z(δ).

Under these assumptions, the first-order optimality conditions with short-live players (in contin-

uous time) are modified in the following straightforward fashion:

h(xt − st)(φ0 + φxx+ φss) +H(xt − st)φx ≤ c′x(xt + δ) + max{0; γx − xt} if xt = −δ or xt = 0,h(xt − st)(φ0 + φxx+ φss) +H(xt − st)φx ≥ c′x(xt + δ) + max{0; γx − xt} if xt = 1,h(xt − st)(φ0 + φxx+ φss) +H(xt − st)φx = c′x(xt + δ) + max{0; γx − xt} otherwise.

and

h(st − xt)(φ0 + φxx+ φss) +H(st − xt)φs ≤ c′s(st + δ) + max{0; γs − st} if st = −δ or st = 0,h(st − xt)(φ0 + φxx+ φss) +H(st − xt)φs ≥ c′s(st + δ) + max{0; γs − st} if st = 1,h(st − xt)(φ0 + φxx+ φss) +H(st − xt)φs = c′s(st + δ) + max{0; γs − st} otherwise,

5.2 Main Result

We have the following straightforward result.

Proposition 4 Suppose that Assumptions 1′, 2′ and 3′ hold. Then Propositions 1 and 3 apply.

Proof. The proof of this proposition is provided in the Appendix.

26

5.3 Comparative Statics

In this subsection, we discuss how changes in parameters affect the steady states and the dynamics

of equilibrium. We focus on the effects of changes in the parameters φx, φs, γx and γs as well as the

cost functions cx and cs. The effects of changes in initial conditions are identical to those already

discussed in Section 3.5

Assumption 3′ guarantees that x∗ = 1 and s∗ = 1 is a steady state. There are also at least two

interior steady states. These steady states are one of two types. The first type is given by x∗ = 0 and

any s∗ that satisfies the following equation:

h(s)(φ0 + φss) +H(s)φs = c′s(δ).

The second type is given by s∗ = 0 and any x∗ that satisfies the following equation

h(x)(φ0 + φxx) +H(x)φx = c′x(δ).

Assumption 3′ guarantees that at least one steady state of each type exists. We impose the following

assumption to make sure that only one steady state of each type exist:

Assumption 4 h(y)(φ0 + φzy) +H(y)φz is a decreasing function of y ≥ 0 for z ∈ {s, x}.

This assumption is fairly mild. The following two conditions would be sufficient to guarantee it:

(i) φz is small, in which case the fact that, from Assumption 3′, h(y) is decreasing for y ≥ 0 ensures

that this assumption is also satisfied, or that (ii) the elasticity of the h function is greater than 1/2,

in which case for any value of φ0, Assumption 4 is satisfied.

Let us focus on the comparative statics of the steady state with x∗ = 0 and s∗ ∈ (γs, 1). The

other case is identical. s∗ solves the following equation:

h(s∗)(φ0 + φss∗) +H(s∗)φs = c′s(δ). (19)

The parameter φx does not directly appear in this equation. Therefore,

∂s∗

∂φx= 0.

Implicitly differentiating with respect to φ0, we get

h′(s∗)∂s∗

∂φ0

(φ0 + φss∗) + h(s∗)

(1 + φs

∂s∗

∂φ0

)+ h(s∗)φs

∂s∗

∂φ0

= 0.

Therefore,∂s∗

∂φ0

=−h(s∗)

h′(s∗)(φ0 + φss∗) + 2h(s∗)φs

> 0,

where the inequality follows from Assumption 4. Implicitly differentiating equation (19) with respect

to φs, we get

h′(s∗)∂s∗

∂φs(φ0 + φss

∗) + h(s∗)

(s∗ + φs

∂s∗

∂φs

)+ h(s∗)

∂s∗

∂φsφs +H(s∗) = 0.

27

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

Figure 6: Changes in steady states and dynamics in response to an increase in φx. The red curvesdepict the boundaries between the basins of attraction of the different steady states when φx = 0 andthe green curves show the same boundaries when φx = 0.1.

Therefore,∂s∗

∂φs=

−h(s∗)s−H(s∗)

h′(s∗)(φ0 + φss∗) + 2h(s∗)φs

> 0,

where again the inequality is a consequence of Assumption 4. Let us next focus on comparative statics

with respect to the cost function. Clearly γs, γs, and cx(·) do not affect the solution of equation (19).

Therefore,∂s∗

∂γs=∂s∗

∂γx= ∇cx(·)s

∗ = 0,

(where ∇cx(·) denotes the (Gateaux) derivative of the steady-state level of state capacity with respect

to the cost function of society).

But the marginal cost of increasing capacity affects the location of the steady state. To quantify

this effect, let us implicitly differentiate equation (19) with respect to c′s(δ):

h′(s∗)∂s∗

∂c′s(δ)(φ0 + φss

∗) + h(s∗)∂s∗

∂c′s(δ)+ h(s∗)

∂s∗

∂c′s(δ)φs = 1.

Therefore,∂s∗

∂c′s(δ)=

1

h′(s∗)(φ0 + φss∗) + 2h(s∗)φs

< 0.

Even though there are unambiguous comparative statics of changes from changes in the output

and cost functions on s∗ and x∗, it has to be borne in mind that these are the values of state and

civil society capacity in a given steady state. The more important conclusion continues to be the

one already highlighted in Proposition 2, that comparative statics in this model are conditional.

Proposition 2 emphasized this for initial conditions, but they are no less true when we consider

28

changes in the output or cost functions. For instance, an increase in the marginal benefit of the

capacity of civil society on output, φx, increases x∗ as we have just shown. However, such a change

also shifts the boundaries of the basins of attraction of the different steady states as depicted in Figure

6. As a result, an economy that was previously in Region II — the basin of attraction of the steady

state (1, 1) — can now shift to the basin of attraction of the corners steady state (0, x∗) in Region III.

Consequently, the long-run state capacity may end up decreasing rather than increasing following an

increase in φx. This reiterates the conclusions of Proposition 2.

5.4 Numerical Results

Figure 6 illustrates how the steady states and the basins of attraction change when we increase φs,

making the capacity of the state more important for overall output. To draw this figure, we use

exactly the same parameterization as in the simulation reported in Figure 4, which corresponds to

the case in which φx = φs = 0 in terms of the model of this section. We then show how the steady

states and dynamics are affected when we increase φx to .1. Particularly noteworthy are the shifts

in the boundaries between the regions, which show that the same type of conditional comparative

statics highlighted in Section 3.5 in response to shifts in initial conditions now apply when we consider

changes in parameters such as the sensitivity of aggregate surplus to the capacity of the state.

In addition, we also show numerically that the same results as those provided above generalize

even when f is concave. Figure 7 depicts the dynamics of state and civil society when we consider

the concave surplus function,

f(x, s) = .6 + 0.1x0.8 + 0.1s0.8.

We can see that in this case, the dynamics are very similar to the ones studied in the section where

the surplus function is linear.

6 Direct Transitions between Region I and Region III

Figure 2 demonstrates how in our main model, the state space is divided into three regions, and

Region II always lies between Regions I and III. However, throughout much of pre-modern history

(though notably not in Ancient Greece which we discuss in the next section), we have many examples

of societies approximating our Regions I and III, but relatively fewer examples of Region II. Perhaps

more challengingly for our model, we observe several transitions from Region I directly into Region

III, which would not be possible in our baseline model, since Region II is in-between and should be

traversed. Here we present a simple modification of the model where Region II shrinks, and creates

a subset of the state space (with low levels of state and civil society strength) where Regions I and

III are adjacent. The basic idea is to modify the model such that the economies of scale in the cost

of investment function becomes dependent on relative strengths.

Suppose that the cost functions for the two players take the form

Cx(xt, xt−∆) = c

(xt − xt−∆

∆+ δ

)+ [max {γ − xt−∆, 0} −max {γ − st−∆, 0}]

(xt − xt−∆

∆+ δ

),

29

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

Figure 7: Dynamics when the aggregate output function, f(x, s), is concave.

and

Cs(st, st−∆) = c

(st − st−∆

∆+ δ

)+ [max {γ − st−∆, 0} −max {γ − xt−∆, 0}]

(st − st−∆

∆+ δ

),

where we have made two changes relative to our baseline model. First, we have made c and γ the

same for the two players, which is just for simplicity’s sake. Second and more important, we have

changed the formulation of economies of scale in conflict, so that it is the relative strength of the two

players that matters. In particular, when both x and s are less than γ, the second term in the cost

function becomes simply a function of the gap between x and s. Clearly this leaves the dynamics

when xt > γ and st > γ unchanged. Consider the case in which xt < γ and st < γ. The differential

equations for the strength of society and state can now be written as

x = (c′)−1(h(x− s) + x− s)− δ

s = (c′)−1(h(s− x) + s− x)− δ.

Therefore, defining a new variable z = x− s, we have

z = (c′)−1(h(z) + z)− (c′)−1(h(z)− z).

Or approximating this around z = 0, we have

z =2z

c′′(δ).

Thus regardless of whether x ≷ s, the gap between these two variables will grow, with either x or s

increasing. Moreover, with x and s sufficiently small, this implies that we converge to one of these

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s

Figure 8: Dynamics with the relative formulation of increasing returns to scale in the cost function forinvesting in capacity for the state and society. We see that shrinking of Region II and the possibilityof direct transitions between Regions I and III.

two variables being zero. Therefore, we can conclude that there exists a neighborhood of (0, 0), such

that starting in this neighborhood, Region II is absent, and the economy will go to either of the

two steady states in Regions I or III. This is depicted in Figure 8, where we use exactly the same

parameterization as in Figure 4, except that we use the cost functions in this section and also set

cx(i) = cs(i) = 9 × i2, and let γx = γs = 0.4. This pattern implies that starting with low values of

state and civil society strength, a society that starts with a weak state could transition directly into

one on the path to a despotic state. However, when we consider societies with sufficiently developed

states and civil societies, transitions from despotic or weak states could take us towards an inclusive

state.

7 Weak, Despotic and Inclusive States in Classical Greece

A revealing example of the divergence of state capacity and institutions is that which took place in

Classical Greece. Existing evidence suggests that at least in the Greek Dark Ages,21 there were few

differences in the institutions of different Greek societies. In the ‘Catalogue of Ships’ in the Iliad,

for example, Homer recounts the forces that united to sail to attack Troy. These involved; Athens;

Sparta (Lacedaemonia); and a collection of cities in the Peloponnese also led by Menelaus the King of

Sparta: Messe, Augeaes, Amyclae and Laas, part of an area known as the Mani. Leaders and people

gathered from all over mainland Greece to join the hunt for Hellen and the subsequent siege of Troy.

21The standard periodization of Greek history is: the Dark Ages, 1200-750 BC, the Archaic Period, 750-480 BC, andthe Classical Age from 480 to 323 BC (Morris, 2010, p. 100).

31

They shared the same language and religion, thought that the Gods lived on Mount Olympus, and

that Zeus was their King. Although there was plenty of rivalry between Achilles and Agamemnon,

they both traced their ancestry back to the mythical ancestor Hellen — as did all Greeks. They all

grew the same crops — the triad of wheat, olives and grapes. They used the same iron agricultural

technology. They lived in lands that were geographically very similar to each other. They had similar

military technologies and used the same tactics such as the famous hoplite warfare. Homer’s poem

does not suggest there were significant cultural, ethnic or institutional differences between the warriors

from Athens, Sparta and the Mani.22

Yet during the Archaic and Classical periods, there was a dramatic divergence in the nature of

their states and societies. On the one hand, many Greek city states, epitomized by Athens, developed

inclusive states. Others, like Sparta, created despotic states. Still others, like those polities inhabiting

the Mani peninsula of the Peloponnese, never developed proper states at all, and indeed fought tooth

and nail against other states and empires extending their influence into the Mani. These outcomes

bear an obvious resemblance to our three steady states.

7.1 Athens: An Inclusive State

In the century after 600 BC Athens developed a centralized state with a great deal of popular involve-

ment and control.23 This state was forged in the context of a conflict between elites and citizens, as in

our model. Aristotle notes that it emerged in the context of “an extended period of discord between

the upper classes and the citizens” (Aristotle, 1996, II.1), while Plutarch discusses the “long-standing

political dispute, with people forming as many different political parties as there were different kinds

of terrain in the country. There were the Men of the Hills, who were the most democratic party, the

Men of the Plain, who were the most oligarchic, and thirdly, the Men of the Coast, who favored an

intermediate, mixed kind of system” (Plutarch, Solon, 13).

For Athens, the pivotal period in the creation of a far stronger state is bracketed by the reforms

of Solon in 594 BC and those of Cleisthenes in 508/07 BC. Aristotle records that there were eleven

constitutions of Athens up to his time (probably around 330 BC), with Solon’s being the fourth

after the mythical ones of the Ionians and Theseus, and Draco’s first written constitution of 621 BC

(The Constitution of Athens, XLI.2). Solon was made archon, one of the chief executive positions in

Athens, for a year in 594 BC, with a mandate to re-configure institutions. Solon himself observed, in

a preserved fragment of his writings, that his institutional design was intended to create a balance of

power between the rich and the poor:

“To the people I gave as much priviledge as was sufficient for them, neither reducing

nor exceeding what was their due. Those who had power and were enviable for their

22Though Homer’s poem is supposedly describing Bronze Age society, the standard view amongst scholars is that itmore accurately represents late Dark Age society with which Homer was familiar. For example, we know Bronze Agepolities were quite bureaucratized with writing and ‘palace economies,’ but this is nowhere mentioned in Homer (seeFinley, 1954).

23All the historical facts we cite here about Athens and Sparta are standard in works on Greek history, see for examplethe surveys of Hall (2013), Ober (2105a), Osborne (2009), and Powell (2016). These treatments rest heavily on classicalwriters such as Diodorus, Plutarch, and Thucydides, as well as more recent archaeological evidence.

32

wealth I took good care not to injure. I stood casting my strong shield around both

parties and allowed neither to triumph unjustly” Solon quoted in Aristotle (1996, XII.1).

Solon increased the strength of both state and society. From the perspective of the state, first in

significance were his judicial reforms. Solon abolished all of Draco’s laws except one, his homicide

law. As Osborne puts it, the homicide law was “not a law about homicide as such, but about the

extent of the family which can pursue vendetta, and the ways in which conditions can be made safe

for the homicide’s return to the community. It is a law which aims to end the situation where any

killing leads the perpetrator to a life of wandering exile” (Osborne, 2009, p. 176).24 Thus Draco’s

‘constitution’ was really a codification of feuding regulations, something more akin to the Albanian

Kanun, than a real constitution.25 Solon’s laws were very different, and Hall (2013) points out an

interesting feature of the homicide law in that it states that guilt is to be judged by a basileis, an

archaic Homeric word for a ruler which Hall compares to “Big Man” in the ethnographic literature

on political institutions. Hall shows how significant it is in terms of state formation that a “feature

of the earliest laws is the appearance of named offices and magistracies in place of the generic term

basileis.” Bureaucracy began to emerge at the same time as Solon’s laws to implement them and

Solon is attributed with creating a systems of courts known as the Dikasteria as well as re-organized

a pre-existing Athenian judicial institution, the Areopagus.

Second in importance was the reform of the executive. Athenian political system had previously

been oligarchic with the choice of the archons being obscure and likely determined by powerful families.

Solon stipulated that there were to be nine archons. He divided the population into four classes based

on their incomes from land and only men from the top two classes could be chosen as archons (chosen

by lot from a list of people nominated by the four traditional ‘tribes’ of Athens). After serving as

archon, which they could do only once and for a year, a man could serve in the Areopagus.

At the same time, Solon’s laws made society stronger by restoring the rights of Athenian citizens.

At the time many had been forced into debt peonage since “all loans were made on the security of

the person of the debtor until the time of Solon” (Aristotle, 1996, II.2). Solon’s laws made enserfing

an Athenian citizen illegal, eliminating debt peonage, and implemented a land reform by the act of

uprooting the boundary markers of fields. Osborne suggests “the boundary markers will ... have

recorded ... the obligation to pay a sixth of the produce, and in uprooting them Solon would have

been freeing the tenants from landowners, giving them the land they owned, and turning Attica into

the land of small farmers which it was in the classical period” (2009, p. 211). He also eliminated

restrictions on movement and location within Athens.

Second, Solon increased popular control over the newly strengthened state. A popular Assembly,

the Ekklesia, pre-dated Solon and he ruled that all Athenian citizens (non-slave, male) could attend

it. Freeing enserfed Athenians was a critical part of making this institution more democratic. Indeed,

as a consequence of serfdom, Aristotle noted that “the mass of the people ... had virtually no share

in any aspect of government” (The Constitution of Athens, II.3) since they lost their citizenship.

However, while democratizing the Assembly, the Archons, as we noted, were elites. The agenda for

24For the text of the law: https://www.atticinscriptions.com/inscription/IGI3/10425Which is why it became notorious for punishing every crime with death, something quite common in codes like the

Kanun.

33

the Assembly was drawn up by a Council of 400, the Boule, equally representing the four Athenian

tribes, whose membership excluded the lowest income class. Though the poorest people were only

represented in the Assembly, Aristotle observes that “These three seem to be the features of Solon’s

constitution which most favored the people: first and greatest, forbidding loans on security of a

person’s body; second, the possibility of a volunteer seeking justice for one who was wronged; third,

and they say that this particularly strengthened the people, appeal to the court” (The Constitution

of Athens, IX.1). Thus he emphasizes that Solon’s constitution of courts, which guaranteed popular

membership of juries and openness to everyone, was a key aspect of citizen’s control of the state since

“when the people have the right to vote in the courts they control the constitution” (The Constitution

of Athens, IX.1-2).

Third, Solon institutionalized social norms that helped to contain elites. The most important

example is the Hubris Law (Ober, 2005, Chapter 5). This forbade any act of hubris, meaning any

behavior aimed at humiliation and intimidation, against any resident of Attica (the broader region

in which Athens lay). Importantly, people could be charged with acting hubristically towards slaves,

who were thus also protected and people were executed for repeated violations of the law.

Solon’s reforms made the state and society stronger, but they did not stop the contest. After he

left power a series of ‘tyrants’, such as Peisistratos, staged coups and took power. Nevertheless, these

tyrants further strengthened some dimensions of the state. For instance, Peisistratos took a series

of measures to integrate Athens with the countryside in Attica. These included the establishment of

rural circuit judges, a system of roads centered on Athens and the introduction of processions linking

Athens with rural sanctuaries as well as the Great Panathenaea festival. Peisistratos also coined the

first Athenian money.

Ultimately, tyranny collapsed and Cleisthenes was brought to power by a mass popular uprising

against his opponents and their Spartan backers. The reforms he implemented also strengthened both

state and society. With respect to the state, he developed an elaborate fiscal system (see Ober, 2015b,

van Wees, 2013, Fawcett, 2016), which levied a poll tax on metics (resident foreigners), and direct

taxes on the wealthy, who had to pay for festivals or for outfitting warships, a variety of customs tolls

and charges, particularly at the port of Piraeus, and taxes on the silver mines of Attica. Second, the

state began to provide an array of public goods, not just security or coinage, but infrastructure in

the forms of walls, roads and bridges, relief for orphans and the disabled, and prisons. Third, the

state became more bureaucratic and was run by various types of functionaries. Aristotle claims that

in the days of Aristides, probably 480-470 BC, there were 700 men working for the state in Attica

and 700 abroad and in addition 500 guards in the docks and 50 on the Acropolis. The Boule had

authority over expenditure decisions and there were a series of boards of magistrates (usually 10)

which implemented policy. Though these were chosen by lot and served annually, they were aided by

professional and state owned slaves.26 Fourthly, he abolished the four tribes that had provided the

people for Solon’s Boule of 400 and replaced it with a new Boule of 500 composed of people chosen

26The Athenian state at this time had no professional state prosecutors or police force. In fact prosecutions forviolations of laws had to be brought by private citizens and guilty verdicts had to be collectively enforced (Lanni, 2016).Gottesman (2014) shows that to be implemented, laws passed by the Assembly had to generate a great deal of consensusmore broadly in society and be implemented by popular force. ‘Popular’ included women, slaves and non-citizens. Onereason this worked so well was the vibrancy of civil society.

34

by lot from ten new political units which replaced the four tribes of Solon’s constitution. These units

were made up of smaller units called demes which were regionally based within Attica. There were

now no class restrictions on membership. The creation of the regional units in itself was a state

building measure that consolidated the initiatives of Peisistratos. Aristotle, for example, notes that

Cleisthenes “made fellow demesmen of those living in each deme so they would not reveal the new

citizen by using a man’s father’s name, but would use his deme in addressing him” (Aristotle, 1996,

XXI.4)). Thus Cleisthenes tried to underpin his new political institutions with new non-kin based

identities.27

Cleisthenes then deepened popular control over the state. Membership of the Boule was restricted

to citizens over the age of 30 and any person could only serve for a year and at most twice in a lifetime.

This implies that most Athenian citizens served at some point in their life. The Boule president was

randomly chosen and served for 24 hours, again giving agenda setting power to any citizen of Athens

irrespective of their wealth or social background. As Aristotle puts it “The people had taken control

of affairs” (Aristotle, 1996, XX.4).

Cleisthenes also formalized the informal institution of ‘ostracism’. Every year the Assembly could

take a vote as to whether or not to ostracize someone. If at least 6,000 people voted in favor of

an ostracism then each citizen got to write the name of a person who they wanted ostracized on a

shard of pottery. Whichever name was written on the most shards was ostracized — banished from

Athens for 10 years. This law, like Solon’s Hubris law, seems to have formalized existing social norms

which were used to discipline elites. Indeed, Aristotle notes about the law “it had been passed by a

suspicion of those in power” (The Constitution of Athens, XXII.3). Even Themistocles, the genius

behind the Athenians victory at Salamis over the Persians, and probably the most powerful man in

Athens at the time, was ostracized for 10 years sometime around 476 BC. Ostracism was used very

sparingly, however, only 15 people were ostracized over the 180 year period when the institution was

in full force, but the threat of ostracism “off the equilibrium path” was a powerful way for citizens to

discipline elites.

The evolution of the Athenian constitution did not stop with Cleisthenes, this was only the sixth of

Aristotle’s eleven constitutions, but it moved steadily towards both greater empowerment of citizens

and a stronger state. This did not happen without conflict. During the Persian wars, the aristocratic

Aeropagus, which Cleisthenes had left alone, took upon itself more power. In response, reforms by

Ephialtes in 462/1 BC stripped it of most of its power. Democracy was overthrown and restored

twice during the Peloponnesian wars, but the trend was towards “increasing power being assumed

by the people. They have made themselves supreme in all fields; they run everything by decrees

of the Ekklesia and by decisions of the Dikasteria (courts) in which the people are supreme” (The

Constitution of Athens, XL.3).

Apart from the formal political institutions and direct measures to strengthen the power of society

relative to elites, there is evidence for social change over time in Athens in the direction of a greater

“public sphere” and also greater social capital (Jones, 1999, Kierstead, 2013). Gottesman (2014, p.

50) discusses the emergence of what he calls “mixed associations” which became institutionalized

after 306 BC, when a right of association emerged “for many groups that before could not express

27An aspect of state building first studied by Weber (1976).

35

their solidarity publicly.” An earlier institutional innovation, which occurred between 353 and 330

BC, was that of “supplication” whereby people had the right to petition the Assembly and ask for

their action on a particular issue. This practice arose earlier but after this time fully one quarter of

Assembly meetings were given over to dealing with supplicants. Gottesman (2014, p. 103) surveying

the existing inscriptions which resulted from these supplications concludes “they appear to involve

only non-citizens” (see also Forsdyke, 2012).

More broadly there is a lot of evidence that state and society evolved together in Athens. The

nature of Athenian democracy did not remain fixed after 508 BC, and neither did the state. As late

as 337 BC it passed the tyranny law (Teegarden, 2013) to provide citizens with another instrument

to fight against despotism. The dynamics of state and society in Classical Athens fit well into the

inclusive state development path in Region II of our model.

7.2 Sparta: A Despotic State

Like Athens, Sparta had a pivotal moment in the emergence of a strong state, the so-called Great

Rhetra initiated by Lycurgus, and probably around the late 8th century BC (see Finley, 1982, on

the dates). Yet the consolidation of the Spartan state came with a very different relationship with

society.

Like Athens’s reforms under Solon, the Great Rhetra took place in a contest over power and

political institutions. Herodotus says (speaking of the period before the Lycurgian reforms) that

Sparta “had the most disorderly state of all the Greeks” (Herodotus, 1.65.2). Thucydides, similarly,

notes that Sparta was in a “state of civil unrest” (Thucydides, 1.18.1) in the same period. Forsdyke

concludes her analysis by noting (2005, p. 292) “It is safest to conclude, therefore, that both conflict

within the elite and tensions between elites and non-elites were driving forces in the development of

the Spartan political system”.

The reforms of Lycurgus definitively organized Spartan society into three groups. Most important

were the Spartan citizens, the homoioi (equals), also known as Spartiates. These were adult males

over the age of 30 who had gone through the system of agoge (literally meaning leading or guidance),

which involved the formation of collective male age groups, starting from the age of seven, for warrior

training. During adulthood they were members of particular messes and to maintain ones rights as

a Spartan citizen, one had to provide a certain amount of food to the mess. The other classes were

the helots, now reduced to state owned serfs, and the perioikoi who were settled in villages and made

manufactured goods for the Spartiates.28

The institutionalization of this class structure went along with a land reform which divided between

the Spartan citizens the land, particularly that of Messenia, and the helots with it, though the helots

were the property of the state and could not be bought or sold by individual citizens. The produce

of these lands is what Spartans used to provide for their mess.

The state was ruled by the two hereditary kings of the Agiad and the Eurypontid families, who had

religious, judicial, and military roles. They communicated with the oracle at Delphi, presided over

various types of legal cases and led the army into battle. The democratic element of the constitution

28Finlay describes them as “free men probably enjoying local self-government” but “were subject to Sparta in militaryand foreign affairs” (1982, p. 25)

36

was the council of five ephors, elected democratically by the citizens. A ephor could serve for one year,

and only once in a lifetime. The ephors monitored the kings and could depose them for misconduct.

There was another council of 28 elders over the age of 60 plus the two kings known as the Gerousia.

Members of the Gerousia were elected for life and usually seem to have consisted of part of the

royal households in addition to the kings themselves. Finally there was the analogy to the Athenian

Assembly, the Apella which was an assembly of all Spartan citizens.

The type of state that emerged from this process was very different from the Athenian state

however. Though the Great Rhetra created a Spartan state which was militarily powerful (holding

back the Persians at Thermopylae and eventually winning the Peloponnesean Wars), it was less strong

than the Athenian state is several obvious ways. First, it provided far fewer public goods. It coined

no money and made no effort to support trade or mercantile activities which instead were actively

discouraged.29 Second, Sparta had neither the type of bureaucracy that Athens built, nor a fiscal

system. Weapons, for example, were procured directly by fiat from perioikoi while the Athenians

taxed rich people to pay for their fleet.30 Sparta did not build fiscal capacity to finance military

power and instead constructed the Peloponnesean league where they went into a coalition with other

states in the Peloponnese so that they provided troops during wartime. As Morris (2010, p. 155)

puts it

“. . . the Spartans chose a low cost way to concentrate coercive power, outsourcing war

rather than building state capacity, then spent the next century and a half quarreling over

whether or accept its limitations or to restructure their society to build state capacity and

overcome them. The two key issues were the relationship between citizens and helots and

oliganthropia, the decline in citizen numbers.”

Morris here emphasizes a third issue, rather than broadening the citizenship base as Solon and

Cleisthenes had done and trying to free it from kinship to build an Athenian identity, the Great

Rhetra narrowed citizenship. This narrowing of citizens was not just about the definition of helots

and perioikoi. Over time increasing concentration of wealth meant that fewer and fewer Spartiates

could actually provide the food required to stay members of their mess. Indeed, Aristotle noted that,

“some of the Spartan citizens have quite small properties, others have very large ones: hence the

land has passed into the hands of a few ... although the country is able to maintain 1500 cavalry and

30,000 hoplites, the whole number of Spartan citizens fell below 1000.”

A key to thinking about why this might be is that the Great Rhetra did not institutionalize the

power of society the way that Solon or Cleisthenes did. First, and most obvious, the vast mass of the

population, probably around 90%, were reduced to hereditary slavery. Athens of course had slaves

as well, but this was around 25% of the population31 and as we noted slaves were protected by the

rule of law and seem to have even played a role in enforcing the law. In contradistinction, the ephors

declared war on the helots every year and Spartiates were even encouraged to murder them. Second,

29There was a type of money consisting of iron bars but it was so unwieldy that it was useless as a medium of exchange.30Aristotle says “the revenues of the state are ill-managed; there is no money in the treasury, althought they are

obliged to carry on great wars, and they are unwilling to pay taxes” The Politics, 1271 10.31This is the estimate of Morris and Powell, (2006, p. 210) for the 4th century BC. It is possible that it was higher

at the time of Cleisthenes.

37

though there was an assembly in Sparta, and though this had to vote on policies formulated by the

kings or the Gerousia, “unlike the Athenian Assembly the Spartans rarely discussed proposals: they

normally only voted (by shouting)” (Morris, 2010, p. 123). The Great Rhetra did not create the

same sorts of democratic institutions which Athens had. Indeed, while the Great Rhetra did endorse

“the final decision-making authority of the demos, it also made it fairly explicit that its role was

primarily ratifying proposals formulated by the kings and the aristocratic council ... the nominal

supremacy of the Spartan demos was little more than an illusion” Hall (2013, p. 220). Leading

scholar of Sparta Hodkinson (1983, p. 280), sums up his view by stating “for all the uniqueness of

the Spartiates upbringing and way of life [the political system] perpetuated the existence of a typical

Greek aristocracy.”

In terms of our model, in Sparta a state emerged which was much more despotic in the way it

treated a society which was far less strong and organized than Athenian society. Again there was a

contest for power, but it evolved dynamically in a very different way. In terms of our model Sparta

fits into Region I.

7.3 The Mani: A Weak State

While Greece experienced different patterns of state formation in the Classical era, not all parts of

it underwent such dynamics of political institutions. Both Thrace and Scythia to the northeast seem

to have not experienced it until much later and instead stayed as stateless societies based on kinship.

Hall (2013, p. 91) distinguishes between different parts of Greece where the polis, the city state,

emerged first, possibly as a consequence of greater cultural and settlement continuity with Mycenean

Greece. In contradistinction to the polis was the ethnos, which he describes as “a group of people -

or more generally, a population. Its common identity resided in the bonds of kinship, however fictive,

that were recognized by its members, bolstered no doubt by shared rituals and customs ... while the

population of a polis take their name from an urban center, those populations that are described as

ethne typically give their name to the general region they inhabit.”

Within the heart of Greece, some ethne stayed on the fringe of the states attempting to maintain

their independence and statelessness. One was the Mani, located on the Tainaron peninsula in the

Southern Peloponnese. Mani was on the borders of Laconia in Sparta and it is probable that during

Sparta’s heyday, its citizens were mostly perioikoi, though probably with some helot communities as

well (Mexis, 2006, pp. 43-45). As we noted, they are described by Homer as having been mustered

by Menelaus at the time of the Trojan War.

Though they may have lived under the shadow of Sparta historically, the Maniates managed to

have subsequently avoided the creation of centralized authority. They maintained this equilibrium not

just in the Classical period of Greece, but also subsequently under Roman, Byzantine, Venetian and

Ottoman rule. None of these empires were able to exert their authority or rule over the Mani. During

the Roman period they were recognized as the ‘Commonwealth of the Lacedaemonians’ and Mexis

notes “on the Tainaron Peninsular Roman colonizers never established themselves” (2006 p. 92).

With respect to the Ottomans, “The armed Maniati populace fought til the last towards two ends.

The first was to keep the Turks out of Mani. The second was for the individual producer-cultivator

himself personally not to have a bond of subservience with the Turkish feudalist. And on both scores

38

they were successful” (Mexis, 2006, p. 352).

In its stead the society was governed in a very decentralized way by clans and kinship groups

who developed a system of conflict resolution based on feuding and vendetta of a type very similar

to the one we described in Montenegro in the Introduction as well as many other stateless societies.

Mexis notes “During the period of Turkish occupation courts did not exist in Mani; neither did

they exist during the period of the national revolutions of 1821. And neither were there any after

independence” (2006, p. 409). Instead clans resolved disputes and also engaged in prolonged conflicts

(many documented in Mexis, 2006, see also Fermor, 1958).

7.4 Comparison

As we mentioned above, during the Greek Dark Ages there seems to have been little to distinguish

Athens, Sparta and Mani, even if they ended up in dramatically different places. Moving closer to

Solon or the Great Rhetra, the similarities are again evident.32

Yet they diverged, and both elites and citizens made different decisions that led to and intensified

that divergence. For example, while the Spartans were consolidating the dependent status of perioikoi,

“the situation in early sixth century Athens may have been fairly close to the situation in contemporary

Laconia” (Hall, 2013, p. 256) to the extent that something close to the perioikoi, the dependent

artisans of Sparta, existed in Attica. Nevertheless, “In the final decade of the sixth century ... the

city made a choice concerning its perioikic neighbors that was not taken by Sparta” (Hall, 2013, p.

257).

Just as in Athens, where the state-building reforms of Cleisthenes were designed to break down

kinship ties and replace them with new identities, the Great Rhetra was designed to “transfer al-

legiance away from the family or kinship group to various male groups ... the family, in sum, was

minimized as a unit of either affection or authority, and replaced by overlapping male groupings” (Fin-

lay, 1982, p. 28). But the lost role of the family was taken over by the agoge, something controlled

by elites.

A final interesting point of comparison and divergence is the differing fates of social norms bol-

stering the control of society over the state. We discussed above Solon’s Hubris law and Cleisthenes’

Ostracism law. There was no such law in Sparta after the Great Rhetra. But the evidence is that

32Most existing scholarly work focuses not on such variation, but on why Classical Greece was so different from BronzeAge (Mycenean Greece) or other parts of the Mediterranean basin or the Near East (such as Persia). Seminal workby Morris (1987, 1996) has argued that the deconcentration of political power after 1200 BC led to the emergence ofmuch more democratic societies where citizens had relatively greater power compared to elites. Morris tracked theconsequences of this for grave goods and the location and number of graves in addition to the distribution of house sizes.Though the

evidence is compelling, it does less well at explaining the variation within Greece. This is also true of other argumentsoften used in this context such as the democratizing impact of iron technology (Childe, 1942, and Snodgrass, 1980) sinceiron was widely used everywhere. Similarly, the argument that the broader access to writing and literacy which camewith the transition from Linear B to Greek, had a democratizing effect (Ober, 2015a) cannot explain the variation weare interested in. A further argument is that elites did not exercise religious power, which while true again does notdistinguish our three cases. Finally, the same applies to the argument that the creation of hoplite warfare redistributedpower towards citizens. Both the Athenians and the Spartans used hoplites, but significantly in the Spartan case,because of the organization of the economy, the military equipment of the hoplites was provided by the state, not bythe individual hoplite (Finlay 1982, p. 30). Here the political and social organization of the state probably trumpedany empowering impact military technology or tactics might have had.

See Hodkinson ed. (2009) for a recent collection of essays by Classical scholars that does focus on the differencesbetween Sparta and other Greek states.

39

such social norms were widespread in Archaic Greece. Forsdyke, for example, argues “In general,

consideration of the evidence from outside Athens suggests that Athenian ostracism was simply one

elaboration of a more generalized Greek practice of using written ballots - whether leaves or potsherds

- as a means of determining a penalty (removal from public office or exile)” (Forsdyke, 2005, p. 285).33

Though Sparta did not use the institution of ostracism in the same way there is evidence of the legacy

of similar institutions which partially lived on even after the Great Rhetra. Forsdyke (2005, Appendix

Three) points out how both Athens and Sparta used exile as a judicial punishment and in Sparta

it was used several times to discipline kings. For example, Leotychidas in 476 BC and Pleistoanax

in 446/5 BC were both exiled for accepting bribes. Given Forsdyke’s arguments about the common

origins of ostracism and exile it seems likely that this use of exile to discipline kings is a residue of

the types of social norms which led to the laws of hubris and ostracism in Athens. That such norms

did not perpetuate themselves after the Great Rhetra may be due to the massive re-organization of

society that this entailed. Thus it is not that Sparta lacked the social norms that the Athenians

had; rather, they were present in a situation where elites were initially more powerful. Hence after

the Great Rhetra, instead of these social norms being institutionalized and thus strengthened, as in

Athens, they were instead significantly eroded, partially surviving only in the judicial instrument of

exile.

Just as there were many similarities between Athens and Sparta, so there were between these

two societies and the Mani. We noted for example that Draco’s Homicide Law was a codification of

feuding regulations suggesting that the institutional equilibrium in Athens at that time was not so

different from that of Mani. Yet Solon’s reforms initiated a transition towards greater state capacity.

This process is in fact captured in Aeschylus’ trilogy The Orestia, which is the classic depiction of

the transition from a stateless society where conflicts are resolved by feuding and revenge (see Finlay,

1954, for a characterization) to one based on law, modelled after Aeschylus’ own Athens. In the first

play, Clytaemnestra murders her husband Agamemnon after his return from the Trojan wars. In

the second play, The Libation Bearers, Agamemnon’s son, Orestes, murders his mother in revenge.

The chorus eggs him on with the words “stroke for bloody stroke be paid, The one who acts must

suffer, Three generations strong the word resounds.” (The Libation Bearers, 315-321). Here Aeschylus

depicts a society based on the feud, on ‘retribution’ and ‘stroke after bloody stroke’, lacking centralized

authority. But in the final play, The Eumenides, Orestes is sent by the god Apollo to Athens pursued

by the Furies seeking revenge for his killing of Clytaemnestra. But in Athens, the patron goddess

Athena breaks the cycle of revenge by creating a court which judges Orestes. The nature of the feud

also seems to be very ancient in Mani society. Mexis points out the Mani vendetta, or ‘chosia’ bares

a strong resemblance to surviving depictions of the Dorian ‘krypteia’:

“Maniati ‘chosia’ has to be a survival of the methodology of a ‘krypteia’ that was

preserved in the traditions of the armed clan” (Mexis, 2006, p. 382).

Moreover, “Such a supposition is supported also by the fact that the vendetta in Mani was a

means for the solution of the differences between the “powerful”. Thus he argues that there are very

33Archaeological evidence attests to ostracism being used as an institution outside of Athens in Argos, Cyrene (inLibya), Megara, Syracuse and Tauric Chersonesus (in the Crimea), see Robinson (2011).

40

deep roots of the vendetta in Mani stretching back to the Classical period.34

According to our theory one can understand the divergence of Athens, Sparta and the Mani as a

consequences of conflict between the state and citizens, but in a situation where there were initially

differences, possibly small ones, in the balance of power between state and citizens. To understand

the divergence of Athens and Sparta, for example, our model suggests one should look for differences

in the balance of power between citizens and elites. A natural interpretation is that social norms

regulating the exercise of power by elites and hierarchy were stronger in Athens than in Sparta, and

even stronger in Mani.

There seem to be a number of reasons for believing that this was indeed the case. Consider first

Athens and Sparta.

First, potentially important is that the Spartans were Dorians who migrated into the Peloponnese

from central Greece at some point in the early Dark Ages. A consequence of this migration was the

enslavement of helots as they expanded out of Laconia where they first settled. Thus one can imagine

the Spartans a bit like colonial societies elsewhere in the world, for example in Latin America after

1492. Coming from the outside, like the Spanish in Latin America, the Spartans found densities of

indigenous peoples and created institutions to exploit them. As everywhere in history, this seems to

have created the basis for hierarchical elite dominated societies. It is possible that the problem of

controlling and exploiting a large population of slaves made the society more militarized and gave

more power to kings and elites who were useful in organizing the militarized suppression of the helots.

Indeed, Rhodes (2011, p. 4) notes that “Originally Sparta’s culture had been like its neighbours,” and

he goes on to argue that one source of Spartan institutional distinctiveness was indeed that because

of “the conquest of Messenia” there was a “need to keep the subject population under control”. Like

Rhodes, Hall (2013, p. 230) notes “Sparta ... had not originally been so distinct from other Greek

poleis. There must have been some sort of turning point”; he proposes that what distinguishes Sparta

was the fact that the territory over which it ruled was “large” (p. 232).35

Second, though the government of Athens before Solon appears to have been oligarchic, Athens

did not have the type of hereditary kings that Sparta did, and elites could not dominate society. That

elites were more powerful in Sparta is attested to in Plutarch’s life of Lycurgus, where he notes that

Soos, supposedly an ancestor of Lycurgus, was the Spartan king who enslaved the helots. Soos’ son,

Eurypon

“appears to have been the first king to relax the excessive absolutism of his fellow-lords,

seeking favor and popularity with the multitude” (Lycurgus, 2).

There was no such “excessive absolutism” to relax in Archaic Athens, nor anything like a king

who could have implemented a mass enslavement.36

34Nevertheless, due to the nature of the society, the Mani in the Classical period are obviously not documented inthe same way that Athens and Sparta are, so the claim for the historical roots of the clan based feuding society, whileplausible, is conjectural.

35In practice the territory of Sparta was about 80% larger than that of Attica.36Plutarch describes another telling incident. The Spartan assembly, known as the Apella, which existed prior to the

Great Rhetra, used its right to sanction the Spartan kings (of which there were two). During the First Messenian Warof 743-722 BC, in the midst of a revolt by the helots, the slave class of Sparta, the kings Polydoros and Theopobosattacked citizens’ rights by promulgating a law such that “if the people should adopt a distorted motion, the senate and

41

Another final potentially important factor is that the Spartan capital never really became an

urban center like Athens, but remained an agglomeration of four, subsequently five, separate villages

(see Cartledge, 2002). This perhaps made the type of mass democratic participation built in the

Athenian state less feasible and harder to organize and again strengthened the power of elites.

Thus the evidence is consistent with the idea that the despotic state emerged in Sparta based on the

exploitation of helots, and to a lesser extent perioikoi, as a consequence of the initial balance of power

favoring elites. This allowed a relatively pro-elite constitution to be constructed by Lycurgus, albeit

with some notable elements of checks and balances. Equally significant was the social organization

and socialization of Spartiates via the agoge which seems to have eliminated the types of resources

the citizens of Athens had available to discipline elites.

What distinguished the Mani from these other cases to which it seems to have been so historically

similar? In our theory this is a consequence of society being more powerful than in the Athenian case.

There were no elites with institutionalized power in Mani like the Spartan kings or even Athenian

oligarchs who filled the positions of Archons prior to their reform under Solon. Rather, there was

a balance of power between different clans and families. Centralization would have involved one

clan dominating the others and as is well documented in ethnographically observed societies, such

a move would have been stringently opposed by other clans who feared being dominated. In some

circumstances, of course, such state formation does take place, often when some clan gains a military

or other advantage over the other. But in many parts of the world a stateless stalemate occurs.

The long persistence of this in Mani is not unusual. For example, in West Africa, at the time of

the scramble for Africa possibly one third of people lived in stateless societies (Curtin, Feierman,

Thompson and Vansina, 1995). Many of these, such as the Tiv in Nigeria studied by Bohannan

(1958), share many features with the Mani or Montenegrins, and in all of these cases, social norms

and practices aimed at limiting political hierarchy and inequality of political power in society appear

to play a central role.

8 Conclusion

There is a great deal of diversity in the nature of states in the world today, in particular in the extent

to which they have capacity to fulfill basic functions, such as raise tax revenues, establish a monopoly

of violence or effectively regulate society. But societies, not just states, also differ enormously. Some

are highly mobilized and organized collectively, with high levels of ‘social capital’ while others are

not. In this paper we have developed a simple model to understand the variation in state capacity,

arguing that states endogenously acquire capacity in a dynamic contest with society. At the heart

of our model is the notion that elites that control states must contest with society (non-elites) for

control over political power, resources and rents. If the state accumulates capacity — what we called

‘strength’ — then this helps it win this contest. But in response society can also accumulate strength,

for example in the form of collective organization and social norms regulating the exercise of power

by elites, and these help it contest against the state.

kings shall have power of adjournment” (Lycurgus, 6) in effect shutting down the Apella if it did something the kingsdidn’t like.

42

We showed that a simple model incorporating this intuition had three very distinct stable steady

states with very different constellations of state society relations. In one steady state, which we called

a despotic state, the state acquired far more strength than society, in a sense dominating it. In the

reverse situation, where society accumulates more strength than the state, we have a weak state.

Finally, and arguably most interestingly, a rough balance of power between state and society leads

to the emergence of an inclusive state. Our model clarifies how the competition between state and

society in this case leads to a type of state with the greatest capacity. Despotic states, because they

can easily dominate society, have less reason to accumulate as much power and capacity.

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Online Appendix

Proof of Proposition 4

Proof of Proposition 4. The proof of this proposition follows directly from the proofs of Proposi-

tions 1 and 3, with only minor changes to Lemma 4, which we provide next, ruling out the stability

of three different types of steady states. We again treat each type separately.

Type 1: x ∈ (0, γx) and s ∈ (0, γs).

The optimality conditions in such a steady state are

h(s− x)(φ0 + φxx+ φss) +H(s− x)φs = c′s(δ) + γs − s

h(x− s)(φ0 + φxx+ φss) +H(x− s)φx = c′x(δ) + γx − x.

Local dynamics are in turn given by

h(s− x)(φ0 + φxx+ φss) +H(s− x)φs = c′s(s+ δ) + γs − s

h(x− s)(φ0 + φxx+ φss) +H(x− s)φx = c′x(x+ δ) + γx − x.

Since the steady-state levels of state and civil society strength are defined by equality conditions

in this case, local dynamics can be determined from the linearized system, with characteristic matrix

given by(1

c′′s (δ) [h′(·)(φ0 + φxx+ φss) + 2h(·)φs + 1] 1c′′s (δ) [−h′(·)(φ0 + φxx+ φss) + h(·)(φx − φs)]

1c′′x(δ) [h′(·)(φ0 + φxx+ φss) + h(·)(φs − φx)] 1

c′′x(δ) [−h′(·)(φ0 + φxx+ φss) + 2h(·)φx + 1]

),

where we wrote h(·) or h′(·) instead of h(s−x) and h′(s−x) in order to save space (and we will adopt

this shorthand whenever we write matrices or long expressions below). From part 2 of Assumption

3′, we can show that the trace of this matrix is positive. In particular, the trace is given by

1

c′′s(δ)[h′(·)(φ0 + φxx+ φss) + 2h(·)φs + 1] +

1

c′′x(δ)[−h′(·)(φ0 + φxx+ φss) + 2h(·)φx + 1].

Using Assumption 3′, this expression is positive if

h′(s− x)(c′′s(δ)− c′′x(δ))(φ0 + φxx+ φss) ≤ (c′′x(δ) + c′′s(δ))(1 + 2h(s− x)(φs + φx)). (20)

Assumption 2′ ensures that

∣∣c′′s(δ)− c′′x(δ)∣∣ ≤ c′′x(δ)(1 + 2h(s− x)(φs + φx))

|h′(s− x)| (φ0 + φx + φs),

which is a sufficient condition for (20), establishing that at least one of the eigenvalues is positive,

and we have asymptotic instability.

Type 2: x ∈ (γx, 1) and s ∈ (0, γs), or x ∈ (0, γx) and s ∈ (γs, 1). Consider the first of these,

h(s− x)(φ0 + φxx+ φss) +H(s− x)φs = c′s(δ) + γs − s

h(x− s)(φ0 + φxx+ φss) +H(x− s)φx = c′x(δ).

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Now, once again, local dynamics can be determined from the linearized system, with characteristic

matrix(1

c′′s (δ) [h′(·)(φ0 + φxx+ φss) + 2h(·)φs + 1] 1c′′s (δ) [−h′(·)(φ0 + φxx+ φss) + h(·)(φx − φs)]

1c′′x(δ) [−h′(·)(φ0 + φxx+ φss) + h(·)(φs − φx)] 1

c′′x(δ) [h′(·)(φ0 + φxx+ φss) + 2h(·)φx]

).

The trace of this matrix can now be computed as

1

c′′s(δ)[h′(s− x)(φ0 + φxx+ φss) + 2hφs + 1]

+1

c′′x(δ)[h′(x− s)(φ0 + φxx+ φss) + 2h(x− s)φx].

which is positive if

h′(s− x)(c′′s(δ)− c′′x(δ))(φ0 + φxx+ φss) ≤ (c′′x(δ) + c′′s(δ))(2h(s− x)(φx + φs)) + c′′x(δ).

The same argument as in the proof of Type 1 establishes that this condition follows from Assumption

2’, and thus at least one of the eigenvalues is positive and the steady state in question is asymptotically

unstable. The argument for the case where x ∈ (0, γx) and s ∈ (γs, 1) is analogous.

Type 3: x = 1 and s < 1 or s = 1 and x < 1.

Let us prove the first case. Such a steady state would require

h(1− s)(φ0 + φx + φss) +H(1− s)φx ≥ c′x(δ)

h(s− 1)(φ0 + φx + φss) +H(s− 1)φs = c′s(δ) + max{0, γs − s}.

We distinguish between s ≤ γs and s > γs. Consider the first one of these. Consider a perturbation

to s + εs for εs > 0 (it is sufficient to consider perturbations that maintain x constant). Then the

local dynamics of s are given by:

s =1

c′′s(δ)[h′(s− 1)(φ0 + φx + φss) + 2h(s− 1)φs + 1]εs.

From Assumption 3′, h′(s− 1) > 0, the conflict capacity of the state locally diverges from this steady

state, establishing asymptotic instability. Consider next the second possibility. In this case, for s+εs,

we have

s =1

c′′s(δ)[h′(s− 1)(φ0 + φx + φss) + 2h(s− 1)φs]εs,

which is also locally asymptotically unstable. The other case is proved identically.

A Model of Economic and Political Investments

In this part of the Appendix, we provide a more detailed model meant to clarify what the strength

of state and society stand for, and show that this model can be mapped to the more reduced-form

set up we use in our main analysis.

Suppose that society consists of a state (ruler) and a number of small producers, each with the

production function

F (gt, kit),

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where gt is a measure of public good provision (such as infrastructure, bureaucratic services or law

enforcement) at time t, and kit designates the capital investment of producer i.

The cost of public good investment by the state depends on what Mann (1986) refers to “infras-

tructural power” of the state, or simply the “presence” of the state, denoted by st. Suppressing time

indices when this causes no confusion, we write this cost as

Γg(g | s).

This dependence captures the fact that investing in public good provision will be much more difficult

for the state when it is not otherwise powerful. There is also a separate cost of increasing the

infrastructural power of the state as specified in the text. In addition, as we discuss below, this

infrastructural power of the state will also determine the state’s relationship with society.

The producers, on the other hand, individually choose their capital level, but also jointly choose

the extent to which they coordinate, which we denote by x. A higher degree of coordination among

the producers might (but need not) impact their costs of investing in capital, which we write as

Γk(k | x),

and this dependence might reflect the fact that a greater degree of coordination among the producers

enables them to help each other or develop greater trust in production relations or internalize some

externalities. More importantly, as discussed in the Introduction, such coordination impacts how

they can deal with the state’s demands. More broadly, this degree of coordination may also stand

for certain social norms that society develops for managing political hierarchy as our historical cases

also emphasize. We assume that the cost of investing in x is as specified in the text.

Note that the assumptions that only s and x, and not g and k, build on their non-depreciated

stock is for simplicity, and facilitate the comparison with our reduced-form model in the text.

The political game takes the following form: first, the state and civil society simultaneously choose

their investments, g and k. Then, the state announces a tax rate τ on the output of the producers.

If the producers accept this tax rate, it is collected and the remainder is kept by the producers. If

they refuse to recognize this tax rate, there will be a conflict between state and society, the outcome

of which will be determined by s and x in a manner similar to the conflict in the text. In particular,

the state will win this conflict if

s− x > σ

and can extract the entire output of producers, while if the inequality is reversed, society wins, and

the state will not be able to collect any taxes. We assume, as in the text, that σ has a distribution

given by the distribution function H.

Here we focus on the economy in discrete time for simplicity and discuss the equilibrium in a single

period. We also suppress time arguments to simplify the notation. The equilibrium can be solved by

backward induction within the period, starting from the tax decision of the state. Given the conflict

technology we have just specified, it is clear that if the tax rate τ is greater than the likelihood of the

state winning the conflict, H(s−x), then there will be a conflict. We may thus focus, without loss of

any generality, on the case in which τ = H(s− x).

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Then the state’s maximization problem can be written as

H(s− x)F (g, k)− Γg(g | s)− Cs(s, s−∆),

where Cs is a cost function for the power of the state similar to the one specified in the text, s−∆

denotes last period’s state strength, and k is the common physical capital investment level of all

agents. The solution to this problem for g can be summarized as

g = g∗(x, k, s).

Note that even though s−∆ influences s, it does not directly impact the choice of g.

Similarly, recalling that 1 −H(s − x) = H(x − s), the maximization problem of citizens can be

written as

H(x− s)F (g, k)− Γk(k | x)− Cx(x, x−∆),

with solution

k = k∗(x, g, s).

Solving this equation together with the equation for g, we can eliminate dependence on the economic

decision of the other party, and obtain an equilibrium (which may not be unique), expressed as

g = g∗∗(x, s),

and

k = k∗∗(x, s).

Substituting these into the payoff functions, we obtain a simplified maximization problem for both

players, essentially replicating our reduced-form model in the text, with the major difference that the

cost functions now also depend on the action of the other player. In particular, the relevant equations

become:

H(s− x)f(x, s)− Cs(s, s−∆ | x),

and

H(x− s)f(x, s)− Cx(x, x−∆ | s),

where

f(x, s) = F (g∗∗(x, s), k∗∗(x, s)),

Cs(s, s−∆ | x) = Γg(g∗∗(x, s) | s) + Cs(s, s−∆)

and

Cx(x, x−∆ | s) = Γk(k∗∗(x, s) | s) + Cx(x, x−∆).

The only complication relative to the model in the text is that because the cost functions depend on

the equilibrium action choices of the other player, there may be non-uniqueness issues, and thus the

relevant statements now will have to be conditional on a particular equilibrium selection.

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Alternative Contest Functions

As noted in the text, the specification of the contest between the state and citizens we have used

so far is not special, and its main properties are shared with general contest functions. To see this,

consider a contest function such that the payoff of the state is

k(s)

k(s) + k(x) + η,

while that of society isk(x)

k(s) + k(x) + η,

where k(·) is an increasing, differentiable function, and η ≥ 0 is a constant. In this case, the marginal

return to increasing investment for the state is (going directly to the continuous time to economize

on space)k′(st)(k(xt) + η)

(k(s) + k(x) + η)2,

and the expression for society is also similar. The cross-partial derivative of this expression, showing

us how this marginal return changes when society increases its investment, is

k′(xt)k′(st)(k(st)− k(xt)− η)

(k(s) + k(x) + η)3.

Notice that when η = 0, this has the same property as our main specification; it is positive when

st > xt, and negative when st < xt. When η > 0, the same result holds provided that st is sufficiently

larger than xt.

A-5